What is a logarithm in simple terms?

A logarithm tells you what exponent you need to put on a base number to get another number. For example, log₂(32) = 5 because 2⁵ = 32.

How are logarithms and exponential functions related?

Logarithms and exponential functions are inverse operations. If y = b^x is an exponential function, then x = log_b(y) is the equivalent logarithmic form, and each undoes the other.

What is a logarithm used for in algebra?

In algebra, logarithms are used to solve equations where the variable is in the exponent, such as 3ˣ = 81. By rewriting this as log₃(81) = x, you can find x using the definition of a logarithm.

Why do some calculators only have log and ln buttons?

On many calculators, "log" means log base 10 and "ln" means the natural logarithm, which is log base e (about 2.718). Other bases can be handled using properties of logarithms or a change-of-base formula, which students may learn in later high school courses.

What Is a Logarithm? Definition and Examples

Kid-friendly explanation

A logarithm answers the question: "How many times do I multiply this number by itself to get that number?" If 2 × 2 × 2 = 8, then log₂(8) = 3 because you used 2 three times. It is like an "undo" button for exponential growth.

Step-by-step examples

Example 1 (simpler): evaluating a logarithm

Find log₁₀(1000).

log₁₀(1000) asks: "10 to what power equals 1000?"

10 × 10 × 10 = 1000, which is 10³ = 1000.

So log₁₀(1000) = 3.

Example 2 (more advanced): solving a logarithmic equation

Solve log₃(x) = 4.

A logarithm and an exponential equation are inverses. The equation log₃(x) = 4 means that 3⁴ = x.

Compute 3⁴: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81. So 3⁴ = 81.

Therefore x = 81.

Going deeper with logarithms

Logarithms are the inverse of exponential functions. If an exponential function has the form y = b^x (with base b > 0 and b ≠ 1), then the related logarithmic function is y = log_b(x). One function "does" the exponential growth, and the other "undoes" it.

On a graph, basic logarithmic functions have a vertical asymptote, which is a vertical line the graph gets closer to but never touches. For the parent function y = log_b(x), the vertical asymptote is x = 0, and the domain (all allowed x-values) is x > 0 because you can only take the logarithm of positive numbers in this context.

Logarithms also appear in logarithmic scales, which are used when values grow or shrink very quickly, such as sound intensity (decibels) or earthquake strength (Richter scale). On a logarithmic scale, equal steps on the scale represent equal ratios (like "10 times bigger"), not equal differences.

Common mistakes with logarithms

Mistake: Treating log_b(x) as b × x, for example saying log₂(8) = 2 × 8 = 16.
Correction: A logarithm is not multiplication; log₂(8) asks for the exponent on 2 that gives 8, so log₂(8) = 3 because 2³ = 8.
Mistake: Ignoring the base and assuming every logarithm is base 10, for example saying log₂(8) = 1 because log(10) is often written without a base.
Correction: The base matters: log₂(8) uses base 2, so you solve 2^? = 8, which gives 3.
Mistake: Allowing nonpositive numbers inside a logarithm in this setting, for example trying to evaluate log₃(−9) or log₅(0).
Correction: For real-number work in middle and high school, the input of a logarithm must be positive, so x > 0 in log_b(x).

How logarithms show up in everyday life

Logarithms are used to describe things that change by large factors. The loudness of sound in decibels, the strength of earthquakes, and the acidity of liquids (pH scale) all use logarithmic scales. This helps compare very big and very small values in a way that is easier to read and work with.

What is a logarithm?

Kid-friendly explanation

Step-by-step examples

Going deeper with logarithms

Common mistakes with logarithms

How logarithms show up in everyday life

Related terms to explore

FAQ about logarithm