Logarithms

Exponents answer "what do I get when I raise b to a power?" Logarithms ask the reverse: "what power do I need?" This inverse relationship makes logarithms the essential tool for solving exponential equations.

Definition

log b (x) = y means b y = x

In words: "the logarithm base b of x is the exponent you put on b to get x."

Reading Logarithms

log₂(8) = 3 because 2³ = 8.

log₅(25) = 2 because 5² = 25.

log₁₀(1000) = 3 because 10³ = 1000.

Common Bases

Notation	Base	Name	Typical Use
log(x)	10	Common logarithm	pH scale, decibels, Richter scale
ln(x)	e ≈ 2.718	Natural logarithm	Continuous growth, calculus
log₂(x)	2	Binary logarithm	Computer science

Properties of Logarithms

These properties follow directly from the exponent rules:

Property	Rule	Example
Product Rule	log_b(MN) = log_b(M) + log_b(N)	log(6) = log(2) + log(3)
Quotient Rule	log_b(M/N) = log_b(M) - log_b(N)	log(5) = log(10) - log(2)
Power Rule	log_b(M^p) = p · log_b(M)	log(8) = 3 log(2)
Identity	log_b(b) = 1	ln(e) = 1
Zero	log_b(1) = 0	log(1) = 0
Inverse	b^log_b(x) = x	10^log(5) = 5

Change of Base Formula

To evaluate a logarithm in any base using your calculator:

log b (x) = log(x) / log(b) = ln(x) / ln(b)

Example: log₃(20) = log(20) / log(3) ≈ 1.3010 / 0.4771 ≈ 2.727.

Solving Exponential Equations

The key strategy: take the logarithm of both sides to bring the exponent down.

Worked Example 1

Solve: 3^x = 81.

Recognize that 81 = 3⁴, so x = 4.

Alternatively: take log of both sides. x · log(3) = log(81), so x = log(81)/log(3) = 4.

Worked Example 2

Solve: 5^2x = 200.

Take the natural log of both sides: ln(5^2x) = ln(200).
Power rule: 2x · ln(5) = ln(200).
Solve: x = ln(200) / (2 ln(5)) ≈ 5.2983 / (2 × 1.6094) ≈ 5.2983 / 3.2189 ≈ 1.645.

Worked Example 3

How long to double an investment at 6% annual interest compounded yearly?

Set up: 2P = P · 1.06^t, so 2 = 1.06^t.
Take log: log(2) = t · log(1.06).
Solve: t = log(2) / log(1.06) ≈ 0.3010 / 0.02531 ≈ 11.9 years.

Common Mistake

log(a + b) is NOT equal to log(a) + log(b). The product rule says log(a · b) = log(a) + log(b). There is no simple rule for the log of a sum.

Practice Problems

Evaluate without a calculator: log₄(64).
Show Solution

4³ = 64, so log₄(64) = 3.
Use properties to expand: log₂(8x³).
Show Solution

log₂(8x³) = log₂(8) + log₂(x³) = 3 + 3 log₂(x).
Solve: 2^x = 50.
Show Solution

x = log(50)/log(2) ≈ 1.6990 / 0.3010 ≈ 5.644.
Condense to a single logarithm: 2 log(x) - log(y) + log(z).
Show Solution

= log(x²) - log(y) + log(z) = log(x²z / y).
A bacteria culture triples every 4 hours. How long until it reaches 10 times its initial size?
Show Solution

Model: N = N₀ · 3^t/4. Set 10 = 3^t/4. Take log: log(10) = (t/4) log(3). So t = 4 log(10)/log(3) = 4 × 1 / 0.4771 ≈ 8.38 hours.

Summary

log_b(x) = y means b^y = x -- logarithms are inverse operations of exponents.
Key properties: product, quotient, and power rules mirror exponent rules.
To solve exponential equations, take the log of both sides and use the power rule.
The change-of-base formula lets you compute any logarithm with a calculator.