Log Calculator — Step-by-Step Solutions

Number

Base

Your calculations will appear here

The Log Calculator computes logarithms with any base. It supports natural logarithms (ln, base e), common logarithms (log base 10), binary logarithms (log base 2), and logarithms with any custom base.

Enter a number and choose your base to get the logarithm instantly. The calculator also shows the relationship between the logarithm and its equivalent exponential form.

How to Use the Log Calculator

Enter the number you want to find the logarithm of.
Select the base: natural log (e), common log (10), binary log (2), or enter a custom base.
The result shows the logarithm value and the equivalent exponential form.

Formula and Theory

A logarithm answers the question: "To what power must the base be raised to produce a given number?" If b^x = y, then log_b(y) = x. Logarithms are the inverse operation of exponentiation.

The three most common logarithms are: the natural logarithm (ln, base e = 2.718...) used in calculus and science, the common logarithm (log, base 10) used in engineering and everyday calculations, and the binary logarithm (log base 2) used in computer science.

Worked Examples

Example 1: Common logarithm

Problem: Find log base 10 of 1000

Solution: 10^? = 1000, so 10^3 = 1000

Answer: log(1000) = 3

Example 2: Natural logarithm

Problem: Find ln(e^5)

Solution: ln(e^5) = 5 * ln(e) = 5 * 1

Answer: ln(e^5) = 5

Example 3: Binary logarithm

Problem: Find log base 2 of 64

Solution: 2^? = 64, so 2^6 = 64

Answer: log2(64) = 6

Frequently Asked Questions

Why is the logarithm of a negative number undefined?

For real-valued logarithms with a positive base, no real exponent can produce a negative result. The logarithm of negative numbers is defined in complex analysis using imaginary numbers.

What is the change of base formula?

log_b(x) = log_a(x) / log_a(b). This lets you calculate any logarithm using just natural or common logarithms.

What are logarithms used for?

Logarithms are used to measure earthquake intensity (Richter scale), sound intensity (decibels), acidity (pH), information content (bits), and to solve exponential growth and decay problems.

What are the key logarithm rules?

Product rule: log(ab) = log(a) + log(b). Quotient rule: log(a/b) = log(a) - log(b). Power rule: log(a^n) = n*log(a).