What is a logarithm?
A power like 2³ = 8 has three parts: the base 2, the exponent 3 and the answer 8. A logarithm asks for the exponent.
logb x = y means by = x. Read it as "log base b of x".
- log₂ 16 = 4, because 2⁴ = 16.
- log₅ 25 = 2, because 5² = 25.
- log₁₀ 0.01 = −2, because 10⁻² = 0.01.
Rules for the parts: the base b must be positive and not 1. The number x must be positive. So log(−5) and log 0 do not exist.
Logs and powers undo each other (they are inverses): blog_b x = x and logb(bx) = x.
Common logs and natural logs
Common log: base 10, written log x (or lg x). It fits our number system: log 1000 = 3, log 1 = 0, log 0.1 = −1.
Natural log: base e ≈ 2.718, written ln x. The number e appears in natural growth, like bacteria or continuous interest. ln e = 1, ln 1 = 0.
Real-life log scales: pH = −log[H⁺], decibels = 10 log(I/I₀), earthquake magnitude. On a semi-log plot one axis goes 1, 10, 100, 1000 in equal steps, so fast (exponential) growth looks like a straight line.
History: John Napier (1614) and Henry Briggs made log tables so that sailors and astronomers could multiply big numbers by adding logs. Slide rules worked on the same idea until calculators came.
The laws of logarithms
For positive a, b and base c (c ≠ 1):
- Product law: log(ab) = log a + log b
- Quotient law: log(a/b) = log a − log b
- Power law: log(aⁿ) = n · log a
- Special values: log_c 1 = 0, log_c c = 1
- Change of base: log_c x = log x ÷ log c = ln x ÷ ln c
Why the product law works: if a = 2³ and b = 2⁴, then ab = 2³⁺⁴. Exponents add when you multiply, and logs are exponents.
Watch out: log(a + b) is NOT log a + log b. There is no simple law for the log of a sum.
The graph of y = log_b x
For b > 1 the graph of y = log_b x:
- exists only for x > 0 (domain), and y can be any real number (range);
- always passes through (1, 0) and (b, 1);
- rises quickly at first, then more and more slowly;
- hugs the y-axis (x = 0 is a vertical asymptote) but never touches it.
For 0 < b < 1 the curve falls instead. The graph of y = log_b x is the mirror image of y = bx in the line y = x, because they are inverse functions.
Shifts work as usual: y = log(x − 3) moves the curve 3 right, so the asymptote becomes x = 3.
Solving exponential and log equations
Exponential equation (unknown in the power), e.g. 5x = 40:
- Take log of both sides: log 5x = log 40.
- Power law: x log 5 = log 40.
- Divide: x = log 40 ÷ log 5 ≈ 1.602 ÷ 0.699 ≈ 2.29.
Log equation, e.g. log₂(x + 1) = 3: rewrite as a power: x + 1 = 2³ = 8, so x = 7. Always check that the number inside each log stays positive.
Inequalities: for base > 1, log is increasing, so log_b u < log_b v means u < v (with u, v > 0).
Modelling: growth A = P(1 + r)t. To find the time t, use t = log(A/P) ÷ log(1 + r).
Key formulas and definitions
- log_b x = y ⇔ b^y = x
- log(ab) = log a + log b
- log(a/b) = log a − log b
- log(aⁿ) = n log a
- log_b x = log x ÷ log b = ln x ÷ ln b
- log_b 1 = 0, log_b b = 1, b^(log_b x) = x
Worked examples
1. Find log₃ 81.
3 × 3 × 3 × 3 = 81, so 3⁴ = 81 and log₃ 81 = 4.
2. Write 10⁻³ = 0.001 in log form.
log₁₀ 0.001 = −3.
3. Simplify log 2 + log 5.
Product law: log(2 × 5) = log 10 = 1.
4. Expand log(x³y/z).
3 log x + log y − log z.
5. Find log₂ 20 with a calculator.
Change of base: log 20 ÷ log 2 = 1.301 ÷ 0.301 ≈ 4.32. Check: 2^4.32 ≈ 20.
6. Solve 2^(x+1) = 48.
(x + 1) log 2 = log 48 → x + 1 = 1.681 ÷ 0.301 ≈ 5.585 → x ≈ 4.585.
7. Solve log x + log(x − 3) = 1.
log[x(x − 3)] = 1 → x² − 3x = 10 → (x − 5)(x + 2) = 0. x = −2 makes log x undefined, so x = 5.
8. ₹10,000 grows at 8% a year. After how many years is it ₹20,000?
20000 = 10000 × 1.08^t → 1.08^t = 2 → t = log 2 ÷ log 1.08 = 0.301 ÷ 0.0334 ≈ 9.0 years.
Common mistakes
- Writing log(a + b) = log a + log b. The product law is about multiplying, not adding.
- Writing log a ÷ log b = log(a ÷ b). The left side is change of base, not the quotient law.
- Keeping answers that make the inside of a log zero or negative. Always check solutions.
- Mixing up log (base 10) and ln (base e) on the calculator.