Log Calculator: Solve Logarithmic Expressions Instantly

Logarithms are essential in mathematics, science, and engineering for solving exponential equations, measuring pH levels, calculating earthquake magnitudes, and analyzing data that spans many orders of magnitude. Our Log Calculator computes logarithms in any base, natural logarithms (ln), and common logarithms (log10), with step-by-step solutions showing how the result is derived.

Understanding Logarithms

A logarithm answers the question: to what exponent must a base be raised to produce a given number? If b^x = y, then log_b(y) = x. For example, log_2(8) = 3 because 2³ = 8. The common logarithm uses base 10, and the natural logarithm uses base e (approximately 2.71828).

Logarithms convert multiplication into addition: log(x × y) = log(x) + log(y). They convert division into subtraction: log(x / y) = log(x) - log(y). And they convert exponents into multiplication: log(x^n) = n × log(x). These properties make logarithms invaluable for simplifying complex calculations.

Using the Log Calculator

Enter the number you want to take the logarithm of, and the base. The calculator supports any positive base other than 1. Select common log (base 10), natural log (base e), or custom base. The calculator shows the result, the logarithmic equation in exponential form, and how the logarithm properties were applied.

You can also solve logarithmic equations: enter an expression like log(x) + log(x+2) = 3 and the calculator solves for x using logarithmic properties.

Applications of Logarithms

pH Scale

pH = -log₁₀[H+], where [H+] is the hydrogen ion concentration. A solution with pH 7 has 10 times more hydrogen ions than pH 8. Each whole pH unit represents a tenfold change in acidity.

Richter Scale

Earthquake magnitude uses a logarithmic scale. A magnitude 6 earthquake releases approximately 32 times more energy than magnitude 5, and 1,000 times more than magnitude 4.

Decibels

Sound intensity is measured in decibels on a logarithmic scale. A 10 dB increase represents a tenfold increase in sound intensity. 90 dB (heavy traffic) is 1,000 times more intense than 60 dB (normal conversation).

Exponential Growth and Decay

Population growth, compound interest, radioactive decay, and drug metabolism all follow exponential patterns. Logarithms are used to linearize these relationships for analysis and prediction.

Logarithmic Properties

• Product rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient rule: log_b(x/y) = log_b(x) - log_b(y)
• Power rule: log_b(x^n) = n × log_b(x)
• Change of base: log_b(x) = log_a(x) / log_a(b) — useful when your calculator only has log₁₀ or ln
• Inverse property: b^(log_b(x)) = x and log_b(b^x) = x
• Log of 1: log_b(1) = 0 for any base b
• Log of base: log_b(b) = 1

Real-World Example

A savings account with $10,000 earns 5% annual interest compounded continuously. How long until the balance reaches $15,000?

• Formula: A = P × e^(rt), so $15,000 = $10,000 × e^(0.05t)
• Divide both sides: 1.5 = e^(0.05t)
• Take natural log: ln(1.5) = 0.05t
• Solve: t = ln(1.5) / 0.05 = 0.4055 / 0.05 = 8.11 years

Checking: log₁₀(1,000) = 3 because 10³ = 1,000. log₂(32) = 5 because 2⁵ = 32. These relationships are the foundation of logarithmic thinking.

Start Calculating

Use our Log Calculator below to solve logarithmic expressions and equations. Also check our Scientific Calculator for advanced calculations and our Standard Deviation Calculator for statistical analysis.