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Logarithm Calculator

Name: Logarithm Calculator
Availability: InStock
Author: Calculat

Calculate logarithms with any arbitrary positive base (log_b x) using the change of base formula with step-by-step mathematical steps.

Argument (x > 0)

Base (b > 0, b ≠ 1)

log_10(100)

Exponential relationship: 10^(2) = 100

Deduction Steps:

Expression: log_10(100)

Change of Base Formula: log_b(x) = ln(x) / ln(b)

ln(100) = 4.60517

ln(10) = 2.302585

log_10(100) = 4.60517 / 2.302585 = 2

Equivalent exponential form: 10^(2) ≈ 100

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What Is a Logarithm Calculator?

A logarithm is the inverse operation to exponentiation. The logarithm of a number x with respect to base b is the exponent to which b must be raised to produce x: log_b(x) = y if and only if b^y = x.

How to Use This Calculator

Enter the argument (x > 0) and the base (b > 0, b ≠ 1).
View the calculated logarithmic value.
Review the change-of-base formula steps and equivalent exponential identity.

Change of Base Formula

\log_b(x) = y \iff b^y = x, \quad \log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}

To compute a logarithm to any base on standard computing hardware, divide the natural logarithm of x by the natural logarithm of base b.

Worked Example

Scenario: Calculate log₂(32)

Given: argument x = 32, base b = 2

By change of base: log₂(32) = ln(32) / ln(2)

Notice that 2⁵ = 32

Result: log₂(32) = 5

Result: log₂(32) = 5

Tips & Key Notes

The argument x must be strictly positive (x > 0); real logarithms of negative numbers and zero are undefined.
The base b must be strictly positive and cannot equal 1 (b > 0, b ≠ 1).
Product rule: log_b(xy) = log_b(x) + log_b(y). Quotient rule: log_b(x/y) = log_b(x) - log_b(y).

Frequently Asked Questions

What is the change of base formula?

The change of base formula allows you to evaluate any base b logarithm using natural log or common log: log_b(x) = ln(x) / ln(b).

Why can’t the base of a logarithm equal 1?

Because 1 raised to any power is always 1 (1^y = 1). Therefore, you cannot solve 1^y = x for any number other than 1.

Why can’t you take the log of a negative number?

In the real number system, raising a positive base b to any real power always yields a positive number (b^y > 0). Negative arguments require complex logarithms.