Practice Maths

L35 — Introduction to Logarithms

Key Terms

Logarithm
logb(x) = y means by = x — the log answers "what power do I raise the base to, in order to get x?"
Base b
The number being raised to a power; must be positive and not equal to 1.
Common logarithm
log10(x), written as log(x) — logarithm with base 10.
Natural logarithm
loge(x), written as ln(x) — logarithm with base e ≈ 2.718.
Log product rule
logb(xy) = logb(x) + logb(y) — the log of a product is the sum of the logs.
Log power rule
logb(xn) = n logb(x) — the log of a power brings the exponent out as a multiplier.

What is a logarithm?

A logarithm answers the question: "What power do I raise the base to, to get this number?"

logb(x) = y  ⇔  by = x

Example: log2(8) = 3 because 23 = 8.

Key logarithm rules

RuleFormulaExample (base 10)
Product rulelogb(mn) = logb(m) + logb(n)log(20) = log(4) + log(5)
Quotient rulelogb(m/n) = logb(m) − logb(n)log(5) = log(10) − log(2)
Power rulelogb(mn) = n ⋅ logb(m)log(103) = 3
Log of 1logb(1) = 0log(1) = 0
Log of baselogb(b) = 1log(10) = 1

Common and natural logarithms

NotationBaseKey use
log(x) or log10(x)10Scientific calculations, pH, decibels
ln(x)e ≈ 2.718Continuous growth/decay, calculus
1 2 3 4 6 8 x −2 −1 0 1 2 3 y (1, 0) (2, 1) (4, 2) (8, 3) y = log₂(x)
y = log2(x): passes through (1, 0), (2, 1), (4, 2), (8, 3). Domain x > 0.
Hot Tip: logb(x) and bx are inverse functions — they undo each other: logb(bx) = x and blogb(x) = x. When stuck, convert between log form and index form.

Worked Example 1 — Converting between log and index form

Write in index form: (a) log3(9) = 2   (b) log10(1000) = 3. Write in log form: (c) 25 = 32.

(a) log3(9) = 2 ⇔ 32 = 9.

(b) log10(1000) = 3 ⇔ 103 = 1000.

(c) 25 = 32 ⇔ log2(32) = 5.

Worked Example 2 — Evaluating logarithms

Evaluate: (a) log2(16)   (b) log10(0.01)   (c) log5(1)

(a) 2? = 16 = 24. So log2(16) = 4.

(b) 10? = 0.01 = 10−2. So log10(0.01) = −2.

(c) log5(1) = 0 (any base raised to 0 is 1).

Worked Example 3 — Log laws

Simplify using log laws: (a) log(3) + log(4)   (b) log(50) − log(5)   (c) log(210)

(a) log(3) + log(4) = log(3 × 4) = log(12).

(b) log(50) − log(5) = log(50/5) = log(10) = 1.

(c) log(210) = 10⋅log(2) ≈ 10 × 0.301 = 3.01.

Worked Example 4 — Solving log equations

Solve: (a) log3(x) = 4   (b) logx(25) = 2

(a) log3(x) = 4 ⇒ x = 34 = 81.

(b) logx(25) = 2 ⇒ x2 = 25 ⇒ x = 5 (x > 0, x ≠ 1).

Worked Example 5 — pH calculation

The pH of a solution is defined as pH = −log10[H+]. Find the pH if [H+] = 10−7 mol/L (pure water).

pH = −log10(10−7) = −(−7) = 7. Pure water is neutral at pH 7.

See Answers ➔

  1. Log and index form. Fluency

    • (a) Write log2(32) = 5 in index form.
    • (b) Write 43 = 64 in log form.
    • (c) Write log10(10 000) = 4 in index form.
    • (d) Write 5−2 = 0.04 in log form.

  2. Evaluating logarithms. Fluency

    • (a) Evaluate log2(64).
    • (b) Evaluate log10(100 000).
    • (c) Evaluate log3(1/9).
    • (d) Evaluate log4(8). (Hint: write 8 as a power of 4.)

  3. Logarithm laws. Fluency

    • (a) Simplify log(6) + log(5).
    • (b) Simplify log(200) − log(20).
    • (c) Simplify 3⋅log(4).
    • (d) Simplify log2(4) + log2(8).

  4. Solving log equations. Fluency

    • (a) Solve log2(x) = 5.
    • (b) Solve logx(81) = 4.
    • (c) Solve log10(x) = −3.
    • (d) Solve log3(x + 1) = 3.

  5. Reading a logarithm graph. Understanding

    The graph above (Key Ideas) shows y = log2(x).

    • (a) State the x-intercept and explain what it means in terms of logarithms.
    • (b) What is the value of log2(8)? Read from the graph and verify algebraically.
    • (c) For what value of x does log2(x) = −1? Verify.
    • (d) Explain why the domain of y = log2(x) is x > 0.

  6. Combining log laws. Understanding

    • (a) Write log(12) in terms of log(2) and log(3).
    • (b) Simplify log2(32) − log2(4).
    • (c) Show that log(25) = 2 − 2⋅log(2).
    • (d) Simplify 2⋅log(3) + log(4) − log(36).

  7. Logarithms in real-world formulas. Understanding

    • (a) pH = −log10[H+]. Find the pH if [H+] = 10−4.
    • (b) Find [H+] if pH = 3.
    • (c) The Richter scale: M = log10(I). An earthquake measures M = 6. How many times more intense is it than M = 4?
    • (d) Decibels: dB = 10⋅log10(I/I0). A sound has intensity 1000I0. Find the decibel level.

  8. Inverse relationship with exponentials. Understanding

    • (a) If f(x) = 2x, find f−1(x).
    • (b) Show that logb(bx) = x.
    • (c) Evaluate log5(57).
    • (d) Evaluate 10log(3.7).

  9. Population doubling time. Problem Solving

    A city’s population grows by 8% each year. Starting at 200 000, we want to know when it will reach 400 000.

    • (a) Write an equation 200 000 × 1.08t = 400 000 and simplify to 1.08t = 2.
    • (b) Take log10 of both sides to get t⋅log(1.08) = log(2).
    • (c) Solve for t and round to the nearest year. (Use log(1.08) ≈ 0.0334, log(2) ≈ 0.301.)
    • (d) The “Rule of 72” estimates doubling time as 72 ÷ (growth rate %). Apply it and compare to (c).

  10. Logarithmic scale. Problem Solving

    Human hearing can detect sounds from 10−12 W/m2 (threshold) to 10 W/m2 (pain).

    • (a) How many orders of magnitude (powers of 10) does this span?
    • (b) Find the decibel level of the pain threshold using dB = 10⋅log10(I/10−12).
    • (c) A sound at 60 dB has intensity I1. A sound at 80 dB has intensity I2. Find I2/I1.
    • (d) Explain why a logarithmic scale is more useful than a linear scale for representing hearing intensity.