Important Note : For real-number logs, the base must be positive and not equal to 1, and the number must be positive.

Use this Log Calculator to solve logarithms with any base. It helps you find the logarithm value, solve for the base, or solve for the number in the equation log_b(x) = y, making it useful for algebra homework, exam prep, and quick math checks.

Reviewed by: AjaxCalculators Editorial Team
Last updated: April 24, 2026
Method source: Standard logarithm definition, logarithmic properties, and change-of-base relationships
Editorial standards: AjaxCalculators Editorial Policy

What This Log Calculator Calculates

This calculator solves the logarithmic relationship:

log_b(x) = y

Where:

• x = number
• b = base
• y = logarithm result

The live tool is flexible because you can change any one of these values and let the calculator solve the others.

How the Log Calculator Works

A logarithm tells you the exponent needed to raise a base to get a number.

log_b(x) = y means:

b^y = x

That is the most important idea behind logarithms: logs are just another way to write exponents.

Example

log₂(32) = 5 because:

2⁵ = 32

How to Solve for Each Variable

1) Solve the Logarithm Value

If the base and number are known, the calculator finds the exponent:

log_b(x) = y

2) Solve the Number

If the base and logarithm are known, the number is:

x = b^y

3) Solve the Base

If the number and logarithm are known, the base is:

b = x^1/y

This follows directly from rewriting the logarithmic equation in exponential form.

Common Log and Natural Log

Two especially common logarithms are:

• Common logarithm: base 10
• Natural logarithm: base e

Common log is often written simply as log(x), while natural log is written as ln(x).

The natural logarithm uses the constant e ≈ 2.71828.

Change of Base Formula

If you want to evaluate a logarithm with a base your calculator does not directly support, you can use the change-of-base rule:

log_b(x) = log(x) / log(b)

or equivalently:

log_b(x) = ln(x) / ln(b)

This is one of the most useful logarithm formulas in practice.

Important Rules and Restrictions

• The base must be positive.
• The base cannot equal 1.
• The number x must be positive for real-number logarithms.
• Logarithms turn multiplication into addition and division into subtraction through standard log rules.

Worked Examples

Example 1: Solve a Logarithm

Solve log₃(81).

Step 1: Ask what power of 3 gives 81.

Step 2: Check powers of 3.
3⁴ = 81

Answer:
log₃(81) = 4

Example 2: Solve the Number

If log₅(x) = 3, find x.

Step 1: Rewrite in exponential form.
x = 5³

Step 2: Calculate.
x = 125

Example 3: Solve the Base

If log_b(64) = 3, find b.

Step 1: Rewrite in exponential form.
b³ = 64

Step 2: Take the cube root.
b = 4

How to Use This Log Calculator

1. Enter the number x.
2. Enter the base b.
3. Enter the logarithm value y if solving for a missing input.
4. Change any value and let the calculator update the others.
5. Use the result to verify logarithm homework or exponent relationships.

How to Interpret the Result

Logarithm result tells you the exponent needed to raise the base to the number.

Base tells you which exponential system the logarithm belongs to.

Number is the final result of raising the base to the logarithm value.

If the entered base is 10, the problem is a common logarithm. If the base is e, it is a natural logarithm.

Practical Uses of a Log Calculator

• check algebra and precalculus homework
• convert between logarithmic and exponential form
• solve for a missing base, number, or logarithm
• work with common logs and natural logs
• use the change-of-base rule for arbitrary-base logs

References

1. OpenStax College Algebra 2e – logarithmic definition and exponential form
2. OpenStax Intermediate Algebra 2e – logarithm properties and change of base
3. LibreTexts – natural and common logarithms

Related Calculators

• Exponent Calculator
• Percentage Calculator
• Pythagorean Theorem Calculator
• Slope Calculator
• Fraction to Percentage Calculator

Disclaimer: This calculator is for educational and general math use. For real-number logarithms, the base must be positive and not equal to 1, and the number must be positive.