Topic Review — Logarithms and Logarithmic Functions

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1. Fluency

   Q1 — Evaluating logarithms

   Evaluate each logarithm without a calculator.

   (a) log₂(32)    (b) log₃(1/9)    (c) log(10 000)    (d) log₅(5⁷)    (e) ln(e³)    (f) log₄(8)

2. Fluency

   Q2 — Converting between exponential and logarithmic form

   Convert each to the other form.

   (a) 2⁸ = 256    (b) 10⁻⁴ = 0.0001    (c) log₇(343) = 3    (d) ln(x) = k

3. Fluency

   Q3 — Applying logarithmic laws

   Write each expression as a single logarithm in simplest form.

   (a) log₂(5) + log₂(8)    (b) 3log(x) − log(y)    (c) 2log₃(6) − log₃(4)    (d) ½log(x⁴y²)

4. Fluency

   Q4 — Key features of a logarithmic graph

   For the function y = 3log₂(x − 2) − 6, state:

   (a) the vertical asymptote    (b) the domain    (c) the x-intercept    (d) the y-intercept (if it exists)    (e) whether the function is increasing or decreasing

5. Understanding

   Q5 — Describing and applying transformations

   Starting from y = log₃(x), describe the transformations and state the new asymptote and domain for each function.

   (a) y = log₃(x + 5) − 2    (b) y = −2log₃(x − 1) + 4

6. Understanding

   Q6 — Finding the equation from a graph

   A logarithmic function of the form y = A log(x) + k passes through the points (1, 5) and (100, −1). Find the values of A and k, and write the equation.

7. Understanding

   Q7 — Solving logarithmic equations

   Solve each equation. Show all working and check your solutions.

   (a) log₄(3x + 1) = 3    (b) log(x²) = 4    (c) log₂(x + 3) = log₂(5x − 9)

8. Understanding

   Q8 — Solving using log laws first

   Solve log₃(x + 6) + log₃(x) = 3. Show all working and state any rejected solutions.

9. Understanding

   Q9 — pH application

   A chemist measures the pH of two solutions:

   Solution A has pH = 3.4    Solution B has pH = 6.2

   (a) Find the [H⁺] concentration of each solution. Give answers in scientific notation to 2 significant figures.
   (b) How many times greater is the [H⁺] concentration in Solution A than in Solution B?

10. Understanding

    Q10 — Exponential equation solved using logarithms

    Solve each equation. Give exact answers and decimal approximations (3 d.p.).

    (a) 4^x = 100    (b) 2 e^3x = 50    (c) 5^{2x − 1} = 30

11. Understanding

    Q11 — Richter scale comparison

    The 1906 San Francisco earthquake had magnitude M = 7.9. The 1989 Loma Prieta earthquake had M = 6.9.

    (a) How many times more intense was the 1906 earthquake than the 1989 earthquake?
    (b) An aftershock was 250 times less intense than the 1989 earthquake. Find its magnitude, correct to 1 decimal place.

12. Problem Solving

    Q12 — Doubling time and half-life

    Challenge. A bacterial culture grows according to N = N₀ e^0.35t, where t is time in hours.

    (a) Find the doubling time (the time for the population to double), correct to 2 decimal places.
    (b) A radioactive isotope decays according to M = M₀ e^−0.0231t. Find its half-life in years, correct to 1 decimal place.
    (c) If the half-life of the isotope is approximately 30 years, what is the decay rate constant k (4 d.p.)?

13. Problem Solving

    Q13 — Finding the equation of a log graph from its sketch

    Challenge. A logarithmic function has the form y = A log_b(x − h). Its graph has vertical asymptote x = 3, passes through (4, 0) and (12, 2).

    (a) Use the point (4, 0) to find h and explain why this fixes h.
    (b) Use the point (12, 2) along with your value of h to find A and b.
    (c) Write the complete equation.

14. Problem Solving

    Q14 — Solving a logarithmic inequality

    Challenge. Solve the inequality log₂(x + 1) + log₂(x − 1) < 3. State the solution as an interval.
15. Problem Solving

    Q15 — Multi-step application: decibels and intensity

    Challenge. The loudness of sound in decibels is L = 10log(I / I₀), where I₀ = 10⁻¹² W/m².

    (a) A rock concert has a loudness of 110 dB. Find its intensity I in W/m².
    (b) A library has a loudness of 40 dB. How many times more intense is the concert than the library?
    (c) Two sounds have intensities I₁ and I₂. Show that the difference in their loudness levels is L₁ − L₂ = 10log(I₁/I₂).
    (d) By how many decibels must you increase a sound to triple its intensity?