Math · College algebra · Concept

Logarithm rules and exponential equations

A logarithm answers “what exponent?”: log_b x = y means bʸ = x. Logarithms turn products into sums, quotients into differences and powers into multiples, which is why rewriting an exponential equation in logarithmic form, or taking the logarithm of both sides, solves it.

A logarithm is an exponent

log₂ 8 = 3 because 2³ = 8. The base b is positive and not 1, and the input x must be positive, since no power of a positive base is zero or negative. ln x means log_e x, and log x usually means log₁₀ x.

{log}_{b} x = y b^{y} = x

The three rules

Each rule comes from an exponent rule: multiplying powers adds exponents, so the log of a product is a sum of logs.

\begin{aligned} {log}_{b} (x y) & = {log}_{b} x + {log}_{b} y \\ {log}_{b} \frac{x}{y} & = {log}_{b} x - {log}_{b} y \\ {log}_{b} (x^{r}) & = r {log}_{b} x \end{aligned}

Change of base

A calculator has ln and log₁₀. To find a logarithm in any other base, divide two natural logarithms: log₂ 10 = ln 10/ln 2 ≈ 3.322.

{log}_{b} x = \frac{ln x}{ln b}

Solving exponential equations

Isolate the power, then rewrite it in logarithmic form, or take ln of both sides and bring the exponent down with the power rule. 5ˣ = 20 gives x = log₅ 20 = ln 20/ln 5 ≈ 1.861.

Solving logarithmic equations

Combine the logarithms into one, rewrite in exponential form and solve. Then check every candidate in the original equation: a value that makes any logarithm’s input zero or negative must be rejected.

Common mistakes

Splitting the log of a sum: ln(x + y) is not ln x + ln y.
Confusing ln(2x) with 2 ln x: ln(2x) = ln 2 + ln x, while 2 ln x = ln(x²).
Dividing inside instead of dividing the logs: ln 20/ln 5 is about 1.861, not ln 4.
Keeping a solution that makes a logarithm’s input negative.

Key terms

Logarithm: The exponent you need on a base to get a number: log_b x = y means bʸ = x. The base must be positive and not 1, and in the real numbers x must be positive.
Natural logarithm: The logarithm with base e, written ln x. It undoes eˣ: ln(eˣ) = x for every x, and e^(ln x) = x for x > 0.
Common logarithm: The logarithm with base 10, written log x, so log 1000 = 3. In this app log means base 10 and ln means base e; the input must be positive.
Change of base: The identity log_b x = ln x / ln b, which computes a logarithm in any base from natural logarithms, or equally from logarithms in any other common base.
Exponential function: A function with the variable in the exponent, such as bˣ with a fixed base b > 0, b ≠ 1. The natural exponential eˣ is its own derivative.

Work through an example

Solve 3 · 5ˣ = 60, exactly and to three decimal places.

Solve an exponential equation with logarithms →

Solve a logarithmic equation and check it →

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