0064-MWALA_LEARN-LOGARITHMS-SOLVING-FOREVER-2

Objectives: 0064-MWALA_LEARN-LOGARITHMS-SOLVING-FOREVER-2

Standard Form & Logarithms – Questions

Questions

  1. Write each of the following numbers in standard form:
    1. 8 419 000
    2. 45.7
    3. 716
    4. 0.000123
    5. 4
    6. 0.005
  2. Determine the decimal numerals for each of the following expressions:
    1. 9.15 × 105
    2. 8 × 10−3
    3. 1.06 × 102
    4. 2.5 × 101
  3. Compute each of the following expressions, giving your answers in standard form:
    1. (8 × 10−3) × (27.5 × 105)
    2. (12.5 × 104) × (8 × 10−7)
    3. (8 × 10−3) / (5 × 105)
    4. (1.728 × 103) / (1.2 × 102)
  4. Given the formula Q = V2 / R, use mathematical tables to calculate Q when:
    1. R = 5 × 101,   V = 2 × 10−1.
    2. R = 4 × 102,   V = 2 × 102.
  5. Find the value of x in each of the following equations:
    1. log5 x = 4
    2. log5(1/125) = −3
    3. log x = 3
  6. Determine the value of x in each of the following equations:
    1. log(x2 + 3x − 44) = 1
    2. log(2x + 1) = 0
  7. Determine the number whose logarithm is:
    1. base 10 is 6.
    2. base 6 is 6.
  8. Given that log x = 8.0524, find log(βˆ›x).
  1. Given that log 2 = 0.30103, log 5 = 0.69897, and log 7 = 0.84510, calculate without using mathematical tables the value of log(35/2).
  2. Without using mathematical tables, determine the value of log 5 – log 8 + 4 log 2.
  3. Simplify each of the following without using mathematical tables or calculators:
    1. (log √10 / log 10) Γ— log 100
    2. log2 176 – log4 11
  4. Use mathematical tables to find the logarithm of each of the following numbers:
    1. 0.8008
    2. 724 079
    3. 0.0002
    4. 23.9
  5. Find the value of x if:
    1. log x = 4.3217
    2. log x = 2.5543
  6. If 3x = 8, find the value of x.
  7. By using mathematical tables evaluate the following expressions:
    1. 40.5 Γ— 300 Γ— 0.008904
    2. 0.632 Γ— 3.456
  8. Use mathematical tables to determine the value of:
    1. (2.09)10
    2. (0.5216)2/3
  9. Compute each of the following expressions by using mathematical tables:
    1. (8.802 Γ— 0.00123) / (0.01252 Γ— 352080)
    2. √(2.16)3 / (0.534 Γ— 3333)
Questions 18–25 (HTML)

Mathematics – Questions 18 to 25

Formatted for the web (responsive Bootstrap).

    Q18

  1. In two concentric circles, if R is the radius of the larger circle and r the radius of the smaller circle, the area of the ring between the two circles is given by A = π(R2 − r2) mm2. Use mathematical tables to find the area of the ring if R = 12.05 mm and r = 10.05 mm. (Take π = 3.142.)

    2. Q19

  2. Use mathematical tables to calculate the value of T from the formula T = 2π √(l/g), given l = 0.825 and g = 9.81 (use π = 3.142).

    3. Q20

  3. Given that 1/u + 1/v = 1/f, use logarithm tables to find u if v = 47.9 cm and f = 10.28 cm.

    4. Q21

  4. By using mathematical tables and the formula v2 = u2 + 2as, calculate the value of s if it is known that u = 18.5, v = 36, and a = 3.8.

    5. Q22

  5. The formula for finding the volume of a right circular cylinder is V = πr2h. Use mathematical tables to find the value of r, given that V = 64.91 cm3 and h = 3.907 cm (use π = 3.142).

    6. Q23

  6. Given that, (4/3)πr3 = 234.5, use mathematical tables to calculate 4πr2 (use π = 3.142).

    7. Q24

  7. Find the value of x, if (log2 x) (−3 + log2 x) = 4.

    8. Q25

  8. Solve for x, given that log9 x + 3 log3 3 = 4.

Answers β€” Questions 1–5 (Standard Form & Logarithms)

Answers (Questions 1–5)

Each answer shows steps, symbol meaning, an everyday example, and a short profitable idea.

Question 1 β€” Write each number in standard form (scientific notation)

What is standard form? A number in standard (scientific) form is written as a Γ— 10n where:

  • a (the mantissa/significand) satisfies 1 ≀ a < 10 (unless the number is 0).
  • 10 is the base (for decimal system).
  • n (the exponent) is an integer that shifts the decimal point: positive n moves it right; negative moves it left.

  1. 8 419 000
    Place decimal after first nonzero digit: 8.419000. Count how many places we moved the decimal from original (originally at end) to between 8 and 4:
    From 8419000. to 8.419000 we moved the decimal 6 places to the left β†’ exponent 6.
    Answer: 8.419 Γ— 106
    Example: City population 8,419,000 ≑ 8.419 Γ— 106 people.
    Profit idea: Build a dashboard that displays national statistics compactly; sell as a reporting widget to media companies that need consistent, easy-to-read presentation of big numbers.
  2. 45.7
    Move the decimal one place left to get 4.57 β†’ exponent 1 because we moved 1 place.
    Answer: 4.57 Γ— 101
    Example: Price = 45.7 USD = 4.57 Γ— 101.
    Profit idea: Use scientific formatting in inventory exports for marketplaces handling mixed units (tens, hundreds, millions) so automated pricing scripts don't misread amounts.
  3. 716
    Move decimal two places left: 7.16 β†’ exponent 2.
    Answer: 7.16 Γ— 102
    Example: 716 meters ≑ 7.16 Γ— 102 m (good for unit scaling in sensor feeds).
    Profit idea: Offer a conversion API for IoT platforms where sensors report in varying magnitudes; formatting to scientific notation avoids overflow and improves aggregation speed.
  4. 0.000123
    Move decimal right until first nonzero digit: 1.23. We moved the decimal 4 places to the right β†’ exponent -4.
    Answer: 1.23 Γ— 10-4
    Example: Concentration 0.000123 g/ml = 1.23 Γ— 10-4 gΒ·ml⁻¹ (common in lab reports).
    Profit idea: Create a lab-results formatting/validation tool for clinics where tiny concentrations must be displayed correctly; sell to local labs which need error-free reporting.
  5. 4
    Decimal is already after the 4 implicitly: 4.0. We moved zero places β†’ exponent 0.
    Answer: 4 Γ— 100
    Example: 4 units β†’ 4 Γ— 100.
    Profit idea: Standard forms keep APIs consistent; package a small library for e-commerce platforms to normalize numeric formats across locales.
  6. 0.005
    Move decimal right until first nonzero 5: 5.0. We moved 3 places β†’ exponent -3.
    Answer: 5 Γ— 10-3 (or 5.0 Γ— 10-3).
    Example: 0.005 m = 5 Γ— 10-3 m (measurements in engineering reports).
    Profit idea: Provide a data-cleaning microservice to scientific e-commerce sellers (chemicals, lab kit suppliers) that normalizes product concentrations for marketplaces.

Question 2 β€” Convert the given a Γ— 10n into ordinary decimal numerals

Rule: moving the decimal n places to the right for positive n, or |n| places to the left for negative n. Pad with zeros as needed.

  1. 9.15 Γ— 105
    Move decimal 5 places to the right: start 9.15 β†’ 915000 (9.15β†’91.5β†’915β†’9150β†’91500β†’915000).
    Answer: 915,000
    Example: 9.15Γ—105 bytes β‰ˆ 915,000 bytes (~0.9 MB).
    Profit idea: Build a file-size normaliser for file transfer services that shows sizes in human-friendly form while storing scientific notation internally for compression/consistency.
  2. 8 Γ— 10-3
    Move decimal 3 places to the left: 8 β†’ 0.008.
    Answer: 0.008
    Example: 8Γ—10-3 A = 0.008 A (small electrical currents).
    Profit idea: Measurement app for electronics hobbyists that converts scientific notation to readable decimals for novice users; monetise via pro features.
  3. 1.06 Γ— 102
    Move decimal 2 places to the right: 1.06 β†’ 106.
    Answer: 106
    Example: 1.06Γ—102 seconds β‰ˆ 106 s (~1 minute 46 s).
  4. 2.5 Γ— 101
    Move decimal 1 place to the right: 2.5 β†’ 25.
    Answer: 25
    Example: 2.5Γ—101 kg = 25 kg.

Question 3 β€” Compute and give answers in standard form

Rules used (scientific notation arithmetic):

  • Multiplication: (a Γ— 10m)(b Γ— 10n) = (aΓ—b) Γ— 10m+n.
  • Division: (a Γ— 10m)/(b Γ— 10n) = (a/b) Γ— 10m-n.
  • After the operation, normalize the mantissa so it lies in [1,10) by shifting factors of 10 into/out of the exponent.

  1. (8 Γ— 10-3) Γ— (27.5 Γ— 105)
    Step 1: multiply mantissas: 8 Γ— 27.5. Do the arithmetic:
    27.5 Γ— 8 = (20 Γ— 8) + (7 Γ— 8) + (0.5 Γ— 8) = 160 + 56 + 4 = 220.
    Step 2: add exponents: -3 + 5 = 2.
    Combine: 220 Γ— 102. Normalize mantissa: 220 = 2.20 Γ— 102, so total = 2.20 Γ— 102+2 = 2.20 Γ— 104.
    Answer: 2.20 Γ— 104 (or 2.2 Γ— 104) which is 22,000.
    Example: Scaling a small sample concentration (Γ—10-3) by a large batch (Γ—105) β€” common when converting lab batch amounts to factory production.
    Profit idea: Build a B2B converter for chemical manufacturers to instantly scale lab recipes to production runs and charge per-conversion or via subscription.
  2. (12.5 Γ— 104) Γ— (8 Γ— 10-7)
    Step 1: mantissas: 12.5 Γ— 8 β†’ multiply:
    12.5 Γ— 8 = (12 Γ— 8) + (0.5 Γ— 8) = 96 + 4 = 100.
    Step 2: exponents: 4 + (-7) = -3.
    Combine: 100 Γ— 10-3. Normalize: 100 = 1.00 Γ— 102, so result = 1.00 Γ— 102-3 = 1.00 Γ— 10-1.
    Answer: 1.0 Γ— 10-1 (or 0.1).
    Example: Combining a large batch factor with a tiny dilution β€” yields a small final quantity (0.1 units).
    Profit idea: A SaaS tool for recipe scaling in cosmetics or food manufacturing to avoid costly unit mistakes when mixing ingredients at scale.
  3. (8 Γ— 10-3) Γ· (5 Γ— 105)
    Step 1: divide mantissas: 8 Γ· 5 = 1.6. (Long division: 5 into 8 β†’ 1 remainder 3 β†’ 30 β†’ 6 β†’ 1.6.)
    Step 2: subtract exponents: -3 - 5 = -8.
    Combine: 1.6 Γ— 10-8 (already normalized).
    Answer: 1.6 Γ— 10-8.
    Example: Tiny sensor current divided by a large scaling factor β€” common in precision electronics.
    Profit idea: Build precision-analytics for microcurrent devices (medical/IoT) that tracks tiny values safely and flags anomalies for a subscription fee.
  4. (1.728 Γ— 103) Γ· (1.2 Γ— 102)
    Step 1: divide mantissas: 1.728 Γ· 1.2.
    Work the division carefully (digit-by-digit): multiply numerator & denominator by 10 β†’ divide 17.28 Γ· 12:
    • 12 Γ— 1 = 12 β†’ remainder 5.28 (17.28 βˆ’ 12 = 5.28)
    • Bring decimal: 5.28 β†’ 12 Γ— 0.4 = 4.8 β†’ remainder 0.48
    • 12 Γ— 0.04 = 0.48 β†’ remainder 0. So total = 1 + 0.4 + 0.04 = 1.44
    Step 2: exponents: 3 βˆ’ 2 = 1.
    Combine: 1.44 Γ— 101 = 14.4.
    Answer: 1.44 Γ— 101 (or 14.4).
    Example: Velocity scaling or unit conversions where one measurement is an order bigger than the other.
    Profit idea: Add a "special case" solver feature to an engineering toolkit that handles normalization and returns human-readable numbers and units for proposals.

Question 4 β€” Given Q = V2 / R, calculate Q in each case

Symbols and meaning:

  • Q β€” the quantity requested. In electrical context this formula is commonly written P = V2/R where P is power (watts). We'll say Q is the power in watts (W).
  • V β€” voltage across resistor (volts, V).
  • R β€” resistance (ohms, Ω).
  • Rule: (a Γ— 10n)2 = a2 Γ— 102n.

  1. R = 5 Γ— 101,   V = 2 Γ— 10-1
    Compute V2: (2 Γ— 10-1)2 = 22 Γ— 10-2 = 4 Γ— 10-2.
    Now divide by R: (4 Γ— 10-2)/(5 Γ— 101).
    Mantissa division: 4 Γ· 5 = 0.8. Exponent subtraction: -2 βˆ’ 1 = -3.
    Combine: 0.8 Γ— 10-3 = 8.0 Γ— 10-4 (normalized).
    Answer: Q = 8.0 Γ— 10-4 W = 0.0008 W.
    Example: A tiny voltage (0.2 V) across a 50 Ξ© resistor dissipates 0.0008 W β€” tiny heating, relevant for low-power sensors.
    Profit idea: Offer energy-audit services for IoT devices β€” identify where inefficient resistor choices cause wasted power; save energy costs for fleets of devices and bill by savings.
  2. R = 4 Γ— 102,   V = 2 Γ— 102
    Compute V2: (2 Γ— 102)2 = 4 Γ— 104.
    Divide by R: (4 Γ— 104)/(4 Γ— 102) = (4/4) Γ— 104βˆ’2 = 1 Γ— 102.
    Answer: Q = 1 Γ— 102 = 100 W.
    Example: 200 V across a 400 Ξ© device dissipates 100 W β€” this is useful when sizing power supplies or heaters.
    Profit idea: Design & sell compact heaters or drying boxes: use the formula to size resistor elements to reach a target wattage (e.g., 100 W). Offer custom builds to small businesses (food dryers, lab equipment).

Question 5 β€” Find x from logarithmic equations

Logarithm reminder:

  • logb(x) = y means by = x. The base b is the number repeatedly multiplied.
  • If the base is not written (just log), it commonly means base 10 (common logarithm) in school contexts.

  1. log5 x = 4
    Rewrite in exponential form: 54 = x.
    Compute 54: 5Γ—5=25, 25Γ—5=125, 125Γ—5=625.
    Answer: x = 625.
    Example: If a quantity grows 5Γ— each period and after 4 periods it reaches x, then initialΓ—5⁴ = x.
    Profit idea: Provide growth-modelling consultancy for small businesses (5Γ— growth per campaign is rare, but logarithms show campaign multipliers); charge per report.
  2. log5(1/125) = βˆ’3
    Interpretation 1 (evaluation): Recognise 125 = 53 so 1/125 = 5-3. Therefore log5(1/125) = -3 β€” this is a true statement (no unknown).
    Interpretation 2 (if the original intent was to find x from log5 x = -3): Convert to exponential form: x = 5-3 = 1/125.
    Answers: The expression equals -3, and if solving log5 x = -3 then x = 1/125.
    Example: Exponential decay: a quantity reduces to 1/125 of its original strength (3 negative powers) β†’ strong attenuation in filters.
    Profit idea: Use log conversions to model attenuation in acoustic or signal businesses; sell consultancy to audio setup or telecom installers to tune signal amplification versus noise.
  3. log x = 3 (implied base 10)
    Convert: 103 = x.
    Compute: 10 Γ— 10 Γ— 10 = 1000.
    Answer: x = 1000.
    Example: If the logarithm of the number of users is 3, the platform has 1000 users.
    Profit idea: Offer quick log-based analytics for founders: turning log-scales (e.g., DAU logs) into plain numbers to help small teams plan capacity and monetisation strategies.
Answers β€” Questions 6–10 (Logs & Applications)

Answers (Questions 6–10)

Detailed solutions, symbol explanations, vivid examples, and practical/ profitable ideas.

Question 6 β€” Solve for x:

Key facts about logarithms and domain:

  • log x = y (base 10 unless another base given) ⇔ 10y = x.
  • Argument of a logarithm (the expression inside log(…)) must be strictly positive (i.e., > 0).

  1. log(x2 + 3x βˆ’ 44) = 1
    Step 1 β€” Convert to exponential form.
    Definition: log A = 1 means 101 = A. So set x2 + 3x βˆ’ 44 = 10.
    Step 2 β€” Move all terms to one side.
    Subtract 10 both sides: x2 + 3x βˆ’ 44 βˆ’ 10 = 0 β‡’ x2 + 3x βˆ’ 54 = 0.
    Step 3 β€” Solve quadratic by factorization (digit-by-digit reasoning).
    We want integers p and q such that pΒ·q = βˆ’54 and p + q = 3. Try factor pairs of 54: (1,54), (2,27), (3,18), (6,9). Check (6,9): 9 βˆ’ 6 = 3 β†’ So take p = 9, q = βˆ’6 (since product must be βˆ’54 and sum +3).
    Thus, x2 + 9x βˆ’ 6x βˆ’ 54 = 0 β†’ group: (x(x+9) βˆ’6(x+9)) = 0 β†’ (x+9)(xβˆ’6) = 0.
    Step 4 β€” Roots
    Set each factor to zero: x+9=0 β‡’ x = βˆ’9 and xβˆ’6=0 β‡’ x = 6.
    Step 5 β€” Check domain (argument > 0).
    Compute the original logarithm argument for each root:
    • For x=6: 62 + 3Β·6 βˆ’ 44 = 36 + 18 βˆ’ 44 = 10 which is > 0.
    • For x=βˆ’9: (βˆ’9)2 + 3(βˆ’9) βˆ’ 44 = 81 βˆ’ 27 βˆ’ 44 = 10 which is > 0.
    Both arguments equal 10 so both are valid solutions (they produce log 10 = 1).
    Final answers: x = 6 and x = βˆ’9.
    Vivid example: Suppose x models a parameter inside a signal-power formula that, after substitution, results in the same positive power (10). Two different physical setups (x=6 and x=βˆ’9) can produce identical measured power β€” this happens when the measured quantity depends on xΒ².
    Profit idea: Offer automated equation-solvers for STEM students: detect domain restrictions and present stepwise checks (like we did) to avoid wrong answers. Charge per-school or via subscription for exam-prep platforms.
  2. log(2x + 1) = 0
    Step 1 β€” Exponential form: 100 = 2x + 1 β‡’ 1 = 2x + 1.
    Step 2 β€” Solve: Subtract 1 both sides: 0 = 2x β‡’ x = 0.
    Step 3 β€” Domain check: For x = 0, argument is 2Β·0 + 1 = 1 > 0, so valid.
    Final answer: x = 0.
    Vivid example: If log of light intensity measured as log(2x+1) is zero, then the raw intensity is 1 unit; solving yields x=0 β€” maybe meaning no extra amplification was applied.
    Profit idea: Create an interactive math widget for online courses that flags invalid log-domain inputs and visually demonstrates why some algebraic roots must be discarded; license to tutoring sites.

Question 7 β€” Determine the number whose logarithm is:

  1. base 10 is 6
    Interpretation: log10(N) = 6 β‡’ 106 = N.
    Compute: 106 = 1,000,000 (one million): multiply 10 by itself 6 times or append 6 zeros to 1.
    Answer: N = 1,000,000.
    Example: If a dataset has log10(population) = 6, the population is one million.
    Profit idea: Provide log-normalization utilities for data pipelines (big numbers to log-scale) β€” sell to analytics teams that compress ranges for visualization and ML.
  2. base 6 is 6
    Interpretation: log6(M) = 6 β‡’ 66 = M.
    Compute step-by-step:
    • 6 Γ— 6 = 36 (that is 62).
    • 36 Γ— 6 = 216 (63).
    • 216 Γ— 6 = 1296 (64).
    • 1296 Γ— 6 = 7776 (65).
    • 7776 Γ— 6 = 46,656 (66).
    Answer: M = 46,656.
    Example: If a process multiplies objects by 6 each stage and you perform 6 stages starting with 1, you end with 46,656 items.
    Profit idea: Use combinatorial-growth calculators (like the 6^6 example) to pitch viral-marketing models that show explosive scale β€” sell strategy reports to SMEs planning referral campaigns.

Question 8 β€” Given log x = 8.0524, find log(βˆ›x)

Rules used:

  • log(xk) = k Β· log x for any real k (logarithm power rule).
  • Cube root: βˆ›x = x1/3.

Compute directly:
log(βˆ›x) = log(x1/3) = (1/3) Β· log x = (1/3) Β· 8.0524.
Digit-by-digit division (8.0524 Γ· 3)
1) 3 goes into 8 β†’ 2 remainder 2 (because 3Γ—2=6; 8βˆ’6=2). 2) Bring down the next digit (decimal part): we move to decimal so place decimal point in result right after 2 β†’ result so far 2. 3) Bring down 0 (from tenths place of 8.0524): we had remainder 2 β†’ becomes 20. 3 into 20 β†’ 6 remainder 2 (3Γ—6=18; 20βˆ’18=2). So next digit = 6 β†’ 2.6
4) Bring down next digit 5 (hundredths place): remainder 2 β†’ 25. 3 into 25 β†’ 8 remainder 1 (3Γ—8=24; 25βˆ’24=1) β†’ next digit = 8 β†’ 2.68
5) Bring down next digit 2 (thousandths place): remainder 1 β†’ 12. 3 into 12 β†’ 4 remainder 0 β†’ next digit = 4 β†’ 2.684
6) Bring down next digit 4 (ten-thousandths place): remainder 0 β†’ 4. 3 into 4 β†’ 1 remainder 1 β†’ next digit = 1 β†’ 2.6841 and remainder 1. This produces a repeating remainder sequence (1 β†’ 10 β†’ 3 β†’ …) so the division continues with repeating 3's after many places.
Exact form: (1/3) Γ— 8.0524 = 2.684133333... (the 3 repeats forever after the 4).
Answer (rounded to 4 decimal places): log(βˆ›x) β‰ˆ 2.6841. (Exact repeating decimal: 2.6841\overline{3} or 2.6841333….)
Example: If x represents the total number of bacteria and log x = 8.0524 (very large), the cube-root-scale log corresponds to measuring the side-length of a 3D culture where volume ∝ x; converting to length scale uses the 1/3 exponent.
Profit idea: Make a microservice that converts growth-model logs into linear or dimensional measures (like turning population logs into lengths or radii for 3D models) and sell to researchers or educational publishers as an API.

Question 9 β€” Given: log 2 = 0.30103, log 5 = 0.69897, log 7 = 0.84510. Find log(35/2)

Log rules applied:

  • log(a/b) = log a βˆ’ log b.
  • log(ab) = log a + log b.

Compute: log(35/2) = log(35) βˆ’ log(2) = log(5Γ—7) βˆ’ log(2) = (log5 + log7) βˆ’ log2.
Plug in given values (digit-by-digit addition/subtraction):
Step A β€” Add log5 + log7: 
 0.69897
+ 0.84510
-----------
 1.54407
Step B β€” Subtract log2 = 0.30103 from 1.54407: 
 1.54407
βˆ’0.30103
---------
 1.24304
Answer: log(35/2) = 1.24304.
Example: If 35/2 represents the ratio of two measured intensities, its log (base 10) = 1.24304 corresponds to about 17.78 in linear scale (10^1.24304 β‰ˆ 17.78) β€” convertable back via the antilog.
Profit idea: Create a fast offline "log table" widget for field engineers who lack internet access; embed common logs (2,3,5,7 etc.) and allow composition β€” sell to labs and remote teams as a tiny paid tool.

Question 10 β€” Evaluate log 5 βˆ’ log 8 + 4 log 2 without tables

Simplify using log identities and powers:

  • log a βˆ’ log b = log(a/b).
  • kΒ·log a = log(ak).
  • Recognize that 8 = 23 so log 8 = log(23) = 3 log 2.

Compute symbolically: 
log5 βˆ’ log8 + 4 log2
= log5 βˆ’ (3 log2) + 4 log2
= log5 + (4βˆ’3) log2
= log5 + log2
= log(5Γ—2)
= log10
Since log10 (base 10) = 1, the expression equals 1. Using the provided numeric logs: 
log2 = 0.30103
log5 = 0.69897
sum = 1.00000
Final answer: log 5 βˆ’ log 8 + 4 log 2 = 1.
Example: The expression arises when converting a product and power of measurement factors into a single decade measure β€” e.g., combined amplification and attenuation that net a factor of 10 (one decade).
Profit idea: Build a small educational plugin that auto-simplifies log expressions and provides the short explanation path we used (identify powers, substitute, collapse to a single log). Package for LMS platforms and sell as an add-on.

Next

Answers β€” Questions 11–15 (Logs & Tables)

Answers (Questions 11–15)

Each question solved with step-by-step work, symbol explanations, a vivid real-life example and a profitable idea.

Question 11 β€” Simplify without tables or calculators

Reminder: Here log denotes logarithm base 10 unless another base is shown. Useful identities used below:

  • log(a^k) = k Β· log a
  • log 10 = 1 because 101 = 10.
  • log 10^m = m (characteristic equals exponent).

  1. (log √10 / log 10) Γ— log 100
    Step A β€” evaluate log √10 : since √10 = 101/2, by power rule log √10 = log(10^{1/2}) = (1/2)Β·log 10 = 1/2 because log 10 = 1.
    Step B β€” denominator log 10 = 1.
    So the fraction = (1/2) / 1 = 1/2.
    Finally log 100 = log(10^2) = 2. Multiply: (1/2) Γ— 2 = 1.
    Answer: 1.
    Example: If you have a quantity measured on a log scale and you take the square-root (half the log) then rescale by a factor that returns the exponent of 100, you end at unity β€” useful when normalizing scales in signal processing.
    Profit idea: Offer a numerical toolsheet (spreadsheet add-on) that simplifies common log identities for lab technicians who convert decibel/Power scales β€” charge a small one-time fee.
  2. log2 176 βˆ’ log4 11
    Step 1 β€” rewrite log4 11 in base 2 using change of base: log411 = log211 / log24. But log24 = 2 because 22=4. So log411 = (1/2)Β·log211.
    Step 2 β€” rewrite log2176. Factor 176 = 16 Γ— 11 = 24 Γ— 11, so log2176 = log2(2^4 Γ— 11) = log22^4 + log211 = 4 + log211.
    Now subtract: (4 + log211) βˆ’ (1/2)Β·log211 = 4 + (1 βˆ’ 1/2)Β·log211 = 4 + (1/2)Β·log211.
    We can rewrite (1/2)·log211 = log2(11^{1/2}) = log2√11. So whole expression = 4 + log2√11 = log2(16·√11).
    Answers (equivalent forms):
    4 + ½·log211 or log2(16√11).
    Example: Interpreting binary-logarithmic scales: if a dataset has a factor 176 and you remove a 4-base-2 scaling then a fractional log term remains β€” useful in algorithms that mix integer powers of two with other factors (e.g., block sizes in file systems).
    Profit idea: Build a diagnostic tool for storage engineers that expresses combined scaling factors as simple binary-log expressions; sell as a plugin to system management suites.

Question 12 β€” Use mathematical tables to find logarithms

Interpretation: We're asked for base-10 logarithms (common logs). In practice mathematical tables give the characteristic (integer part) and mantissa (decimal part). Below are accurate values (mantissa rounded to 5 decimal places where appropriate).

Numberlog10 (value)Work / comment
0.8008-0.09648Because log(0.8008) β‰ˆ -0.0964759 β†’ round to -0.09648.
724 0795.85979724079 = 7.24079 Γ— 105 β†’ log = 5 + log(7.24079) β‰ˆ 5.85978595 β†’ 5.85979.
0.0002-3.698970.0002 = 2 Γ— 10-4 β†’ log = -4 + log2 β‰ˆ -4 + 0.30103 = -3.69897 (exact to 5 dp).
23.91.3784023.9 = 2.39 Γ— 101 β†’ log = 1 + log(2.39) β‰ˆ 1.37839790 β†’ 1.37840.
Final values (5 dp rounding): log 0.8008 = -0.09648, log 724079 = 5.85979, log 0.0002 = -3.69897, log 23.9 = 1.37840.
Example: Engineers use such logs when converting large sensor arrays to decibel or order-of-magnitude scales; precise mantissa values are needed for combining products and quotients.
Profit idea: Produce a printable 'cheat-sheet' poster of common logs and important values for labs and classrooms; sell via an online store to schools and tutors.

Question 13 β€” Find x when log x = given values

Rule: log x = y implies x = 10^y (since logs are base 10 here).

  1. log x = 4.3217
    Compute x = 10^{4.3217}. Using exponentiation gives x β‰ˆ 20,974.90488 (rounded to 5 significant figures: 20,974.9).
    Answer: x β‰ˆ 20,974.905 (β‰ˆ 2.09749 Γ— 104).
    Example: If a sound level reading's log value is 4.3217, the linear measure is β‰ˆ 20,975 units β€” useful when converting between decibel-like logs and linear intensities.
    Profit idea: Offer conversion widgets for scientists to convert logged dataset values back to linear scale for reports; charge per API call or monthly subscription.
  2. log x = 2.5543
    Compute x = 10^{2.5543} β‰ˆ 358.3438868. Rounded to 6 significant figures: 358.344.
    Answer: x β‰ˆ 358.344 (β‰ˆ 3.58344 Γ— 102).
    Example: Converting logged bacterial counts in a lab back to actual colony counts for reporting.
    Profit idea: Develop a lab-report generator that takes logged measurement inputs and outputs formatted tables and charts for regulatory submissions.

Question 14 β€” If 3^{x} = 8, find x

Method: take logarithms (any base) of both sides and use change-of-base if desired. We'll use natural logarithm ln and the identity ln a^b = b ln a.

ln(3^x) = ln 8 β†’ xΒ·ln 3 = ln 8 β†’ x = ln 8 / ln 3.
Compute values: ln 8 = ln(2^3) = 3 ln 2 β‰ˆ 3 Γ— 0.693147 = 2.0794415. ln 3 β‰ˆ 1.0986123. Division: x β‰ˆ 2.0794415 / 1.0986123 β‰ˆ 1.89278926.
Answer: x β‰ˆ 1.89278926 (exact form ln 8 / ln 3 or log 8 / log 3).
Example: If you have a process that triples every unit time and you want it to reach 8Γ— initial size, it needs β‰ˆ 1.893 time units (non-integer because growth is discrete multiplicative per unit).
Profit idea: Build a growth forecasting microservice for startups modelling multiplicative growth and required time to reach specific multipliers; sell projections and visualisations to early-stage founders.

Question 15 β€” Use tables (or arithmetic) to evaluate

We'll compute directly and show how logs/tables could have been used: log(ab) = log a + log b, and log(a/b) = log a βˆ’ log b. Using direct multiplication is fine for clarity.

  1. 40.5 Γ— 300 Γ— 0.008904
    Compute step-by-step (digit-by-digit):
    First 40.5 Γ— 300 = 40.5 Γ— 3 Γ— 100 = 121.5 Γ— 100 = 12,150.
    Now multiply 12,150 Γ— 0.008904. We can do 12,150 Γ— 0.008904 = 12,150 Γ— (8.904 Γ— 10-3) = (12,150 Γ— 8.904) Γ— 10-3.
    Compute 12,150 Γ— 8.904 carefully:
    • 12,150 Γ— 8 = 97,200
    • 12,150 Γ— 0.9 = 10,935.0
    • 12,150 Γ— 0.004 = 48.6
    • Sum = 97,200 + 10,935 + 48.6 = 108,183.6
    Now multiply by 10-3: 108,183.6 Γ— 10-3 = 108.1836.
    Answer: 108.1836.
    Example: Compute total energy: 40.5 units per item Γ— 300 items Γ— conversion factor 0.008904 β†’ 108.1836 total units β€” typical in batch cost calculations.
    Profit idea: Create an operations-cost calculator for manufacturers to estimate batch energy or material usage quickly; monetise as an add-on for small factories.
  2. 0.632 Γ— 3.456
    Multiply exactly (digit-by-digit):
    • 0.632 Γ— 3 = 1.896
    • 0.632 Γ— 0.4 = 0.2528
    • 0.632 Γ— 0.05 = 0.0316
    • 0.632 Γ— 0.006 = 0.003792
    • Sum = 1.896 + 0.2528 + 0.0316 + 0.003792 = 2.184192
    Answer: 2.184192.
    Example: Multiplying concentration by volume to get mass in laboratory mixing.
    Profit idea: A microservice for lab technicians that offers exact digit-by-digit multiplication and rounding rules to avoid small rounding errors in regulated reports (sell to clinics/labs).

Reference Book: N/A

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