Logarithm Calculator

Evaluate, expand, condense, or solve logarithms while keeping every domain restriction visible.

Drag & drop or upload an image or PDF

Enter a logarithm, such as log_2(x - 1) = 3

Try an example

A logarithm asks for an exponent

The statement log base b of a equals c means b raised to c equals a. This definition is the bridge between logarithmic and exponential form.

Log properties can turn products into sums, quotients into differences, and powers into coefficients. They do not split a logarithm of a sum. Every real logarithm also requires a positive argument.

How to use the logarithm calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Logarithmic and exponential form

For real logarithms, b must be positive and not equal to 1, and a must be positive.

lo g_{b} (a) = c ⟺ b^{c} = a

A reliable way to work through it

Rewrite in exponential form

This is often the shortest path when a logarithm equals a constant.

Use log properties

Expand products and powers, or condense compatible terms into one logarithm.

Check the domain

Reject any candidate that makes a logarithm argument zero or negative.

Worked example

Solve log₂(x - 1) = 3

x - 1 = 2^{3}

Rewrite the logarithm in exponential form.

x - 1 = 8

Evaluate the power.

x = 9

Add 1 to both sides.

x - 1 = 8 > 0

Check the logarithm domain.

The valid solution is x = 9.

Common mistakes to check

Splitting a sum

log(a + b) is not equal to log(a) + log(b).

Keeping an extraneous answer

A solved candidate is invalid if any original log argument is not positive.

Mixing log bases

Product and quotient properties require logarithms with the same base.

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Questions students ask

Can the calculator use any logarithm base?

Yes. Write the base explicitly, such as log_2(32), or use log for base 10 and ln for base e.

Can it expand and condense logarithms?

Yes. State which form you want and it will apply the product, quotient, and power properties.

Why are some logarithm solutions rejected?

Real logarithms only accept positive arguments, so algebraic candidates must be checked in the original equation.

How does the change-of-base formula work?

It rewrites log base b of x as ln(x)/ln(b), or as the same quotient using any other valid base.