Logarithm Equation Calculator

Our free online logarithm equation calculator solves equations of the form \( \log_b(x) = y \) for any unknown variable. Enter any two values among the base (b), argument (x), and result (y) to instantly compute the third with step-by-step solutions. Explore natural logarithms with our Natural Log calculator, or solve exponential equations using the Solve for Exponents Calculator.

What is a Logarithmic Equation?

A logarithmic equation is an equation that involves a logarithm with a variable argument, base, or result. The standard form is:

\[ \log_b(x) = y \]

This is equivalent to the exponential form:

\[ b^y = x \]

Where b is the base (b > 0, b \neq 1), x is the argument (x > 0), and y is the exponent or logarithm value.

How to Solve Logarithmic Equations

Solving for y (the Logarithm Value)

Given \( \log_b(x) = y \), to find y when you know b and x:

\[ y = \frac{\log(x)}{\log(b)} \]

Using the change of base formula, you convert to any base and divide the logs.

Solving for x (the Argument)

Given \( \log_b(x) = y \), to find x when you know b and y:

\[ x = b^y \]

Simply raise the base b to the power y to get the argument.

Solving for b (the Base)

Given \( \log_b(x) = y \), to find b when you know x and y:

\[ b = \sqrt[y]{x} = x^{1/y} \]

Take the y-th root of x to find the base.

Examples

Example 1: Solve for y

Given \( \log_3(5) = y \):

\[ y = \frac{\log(5)}{\log(3)} = \frac{0.69897}{0.47712} \approx 1.46497 \]

So \( \log_3(5) \approx 1.46497 \), meaning \( 3^{1.46497} \approx 5 \).

Example 2: Solve for x

Given \( \log_4(x) = 2 \):

\[ x = 4^2 = 16 \]

So \( \log_4(16) = 2 \).

Example 3: Solve for b

Given \( \log_b(16) = 2 \):

\[ b = \sqrt[2]{16} = 4 \]

So \( \log_4(16) = 2 \).

Properties of Logarithms

• \( \log_b(1) = 0 \) for any valid base
• \( \log_b(b) = 1 \) for any valid base
• \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• \( \log_b(x/y) = \log_b(x) - \log_b(y) \)
• \( \log_b(x^n) = n \cdot \log_b(x) \)
• \( b^{\log_b(x)} = x \)