Practice Maths

T3T3 Review — Indices and Logarithms

Mixed questions covering Index Laws and Scientific Notation (L34), Introduction to Logarithms (L35), and Solving Exponential Equations (L36).

  1. Index laws. Fluency

    • (a) Simplify (2x3)4.
    • (b) Simplify a5 × a−3.
    • (c) Evaluate 813/4.
    • (d) Simplify (x2y3)2 ÷ (x3y).

  2. Scientific notation. Fluency

    • (a) Write 0.000 000 45 in scientific notation.
    • (b) Write 3.06 × 105 as an ordinary number.
    • (c) Calculate (5 × 106) × (3 × 10−2). Give answer in scientific notation.
    • (d) Calculate (8.4 × 107) ÷ (2.1 × 103). Give answer in scientific notation.

  3. Logarithms. Fluency

    • (a) Evaluate log2(128).
    • (b) Write 72 = 49 in log form.
    • (c) Simplify log(8) + log(125) using log laws.
    • (d) Solve log3(x) = −2.

  4. Solving exponential equations. Fluency

    • (a) Solve 23x = 64.
    • (b) Solve 7x = 20. Give answer to 2 d.p.
    • (c) Solve 1.04t = 2. Give answer to 2 d.p. (This is the doubling time at 4% growth.)
    • (d) Solve 0.9n = 0.5. Give answer to 2 d.p.

  5. Interpreting exponential graphs. Understanding

    0 1 2 3 4 5 t 0 40 80 120 160 A (0, 100) (0, 160) A B
    Curves A and B show two different exponential functions over time t.
    • (a) State the initial value of each curve.
    • (b) Which curve represents growth and which represents decay?
    • (c) Estimate the value of curve B when t = 3.
    • (d) For curve A, the growth rate is 15% per year. Write the equation A(t) = 100 × 1.15t and find when A = 200.

  6. Combined index and log skills. Understanding

    • (a) Simplify log2(84) without a calculator.
    • (b) Express √(a3b) using fractional indices.
    • (c) Solve 2x × 4x+1 = 82.
    • (d) Simplify 2⋅log(5) + log(4). (Answer should be a single integer.)

  7. Real-world indices and logs. Understanding

    • (a) A virus spreads so that the number of infected people triples every day. Starting at 2 people, write a formula for the number infected after d days.
    • (b) How many days until there are over 1 000 infected people?
    • (c) The mass of a proton is 1.67 × 10−27 kg and the mass of an electron is 9.11 × 10−31 kg. How many times heavier is a proton? Express in scientific notation.
    • (d) An earthquake of magnitude 7.5 releases 107.5 units of energy. One of magnitude 5.5 releases 105.5 units. How many times more energy does the larger quake release?

  8. Financial applications. Understanding

    • (a) $10 000 is invested at 7% p.a. compounded annually. Find its value after 8 years.
    • (b) How many years until it doubles? (Use log method.)
    • (c) A debt of $5 000 grows at 18% p.a. compounded monthly. Find the amount owed after 2 years.
    • (d) A phone worth $1 200 depreciates at 20% per year. When does its value fall below $200?

  9. Logarithm problem solving. Problem Solving

    • (a) If log2(x) + log2(x + 2) = 3, find x. (Hint: combine logs, then solve the resulting quadratic.)
    • (b) If log(x) + log(x − 3) = 1, find x.
    • (c) Prove that logb(x) = log(x)/log(b) by letting logb(x) = y and converting.
    • (d) Use the change-of-base formula to evaluate log3(20) to 3 decimal places.

  10. Exponential modelling. Problem Solving

    The number of downloads of an app follows N(t) = 500 × 2t/4, where t is the number of weeks since launch.

    • (a) Find the initial number of downloads and the number after 4 weeks.
    • (b) Find how long it takes to reach 8 000 downloads.
    • (c) By what factor does the number multiply every 8 weeks?
    • (d) A second app starts with 2 000 downloads and grows at 10% per week: M(t) = 2000 × 1.10t. Find when both apps have the same number of downloads. (Set up the equation and solve numerically by trying t = 10, 15, 20.)

  11. Scientific measurement. Problem Solving

    • (a) The diameter of a hydrogen atom is about 1.06 × 10−10 m. The diameter of the observable universe is about 8.8 × 1026 m. How many hydrogen atoms laid end-to-end would span the universe? Give your answer in scientific notation.
    • (b) Express the ratio as a power of 10 and find its order of magnitude.
    • (c) A nanometre (nm) = 10−9 m. A red blood cell is about 8 μm (micrometres) in diameter. Write 8 μm in nm. (1 μm = 10−6 m.)
    • (d) How many times larger in diameter is a red blood cell compared to a hydrogen atom?

  12. Connecting indices, logs, and equations. Problem Solving

    • (a) Show that n = log(A/A0) / log(r) gives the time for A0 × rn = A.
    • (b) A population of 3 000 grows at 6% p.a. Use the formula from (a) to find when it exceeds 12 000.
    • (c) A savings account has $P and grows to $2P at 5% p.a. Show that the doubling time is independent of P.
    • (d) Using the same formula, show that the doubling time at rate r% is approximately 70/r years (the “Rule of 70”). (Hint: use ln(2) ≈ 0.693 and ln(1 + r/100) ≈ r/100 for small r.)
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