Log Calculator – Any Base, ln, log₁₀ & log₂

A logarithm tells you the exponent. If b^y = x, then log_b(x) = y. The log calculator on this page evaluates that for any valid base, including log₁₀, ln, and log₂, and it can run the formula backwards.

1. Choose Evaluate log, Solve for x, or Solve for base.
2. Enter the argument and base, or the values you already know for a reverse solve.
3. Use log10, ln, or log2, or type a custom base greater than 0 and not equal to 1.
4. Read the result, the power check, and the change-of-base steps.

Any log can be written with natural logs:

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

The reverse formulas are the ones behind the other two modes:

x = b^y

b = x^(1/y)

y cannot be 0 when you are solving for the base. After you have a power to check, use the Exponent Calculator.

• log₁₀, the common log, counts powers of ten. log₁₀(1000) = 3.
• ln is the natural log calculator preset, base e. ln(e) = 1 and ln(e²) = 2.
• log₂ is the base-2 log. log₂(1024) = 10. There is no separate log base 2 page; the preset is on this calculator.
• Custom base accepts any base greater than 0 except 1, including a base between 0 and 1.

• Exponential equations, where the unknown is in the exponent.
• pH, which is a negative common log of hydrogen ion concentration.
• Decibels, which compare intensity on a log scale.
• Growth and decay, including doubling time and half-life.
• Compound growth, where a log turns a power back into the number of periods.
• Computer science, where log₂ counts how many times a quantity doubles or how deep a binary search goes.
• Other logarithmic scales, such as earthquake magnitude and sound level.

Roots are the matching inverse for a known exponent. Use the Square Root Calculator for square, cube, and nth roots, or the Scientific Calculator when you also need trig or a full keypad.

• Product rule: log(xy) = log(x) + log(y), same base.
• Quotient rule: log(x/y) = log(x) − log(y).
• Power rule: log(x^n) = n · log(x). That is why ln(e²) = 2.
• Change of base: log_b(x) = ln(x) / ln(b). Any valid base works in that formula.
• log_b(1) = 0, because b^0 = 1.
• log_b(b) = 1, because b^1 = b.
• log(0) is undefined. No real power equals 0.
• A negative argument is not a real logarithm.
• log(a + b) is not log(a) + log(b).
• ln and log are not the same key. ln is base e. Unmarked log often means base 10.
• Base 1 is invalid. A base between 0 and 1 is valid.

• Exponent Calculator to check b^y after you find a log.
• Square Root Calculator for roots when the exponent is known.
• Scientific Calculator for a keypad with log and ln keys.
• Calculator Tools for the rest of the math and everyday calculators.