Logarithm: Definition and Example

Definition: A logarithm answers the question: to what exponent must a base be raised to produce a given number? In symbols, log base b of x equals y means b to the y equals x.

What does a logarithm mean?

The statement log_b(x) = y is another way to write b^y = x. The base b must be positive and not 1, and x must be positive. A logarithm is simply an exponent in disguise.

Which rules do you need?

• log_b(MN) = log_b(M) + log_b(N)
• log_b(M/N) = log_b(M) – log_b(N)
• log_b(M^k) = k log_b(M)

Worked example: find log₃(243). Ask which power of 3 equals 243. Since 3⁵ = 243, the answer is 5.

Another: log₂(40) – log₂(5) = log₂(8) = 3.

What mistakes should Mathbot warn you about?

The log of a sum is not the sum of logs: log(A + B) does not equal log A + log B. Also, you can never take the log of zero or a negative number in the real number system.

Where to learn it

• Logarithms: The Inverse of Exponents – lesson
• Using Logs to Solve Exponent Equations – lesson

Related terms

• Natural logarithm
• Change of base formula
• Inverse function
• Exponential equation

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