Logarithm of 125 to base 5

Find the value of the logarithm base 5 of 125, and verify your answer using exponential form.

Step-by-step

1. Base a = 5, argument N = 125
   For logarithms, the base must be greater than zero and not equal to one, and the argument must be greater than zero.
2. log_5 125 = 3
   A logarithm is the inverse of exponentiation; it asks what power of the base equals the argument.
3. Check: 5^3 = 125
   Substitute the result back into exponential form to quickly catch calculation errors.
4. Domain check: base 5 is greater than zero and not equal to one, argument 125 is greater than zero
   The base of a logarithm must be greater than zero and not equal to one; the argument must be greater than zero. Otherwise it is undefined.

Why it works

The argument of this problem is an integer power of the base, so you can find the answer simply by observing the exponent directly. The method is to express the argument as a power of the base; that exponent is the logarithm value. If it cannot be expressed directly, use the change-of-base formula to convert it to common logarithms (base 10). After calculating, you must verify by substitution: plug the logarithm value back as an exponent and check whether it equals the original argument. This step immediately catches calculation errors.

Common mistakes

• Writing the wrong exponent by one digit when expressing the argument as a power of the base.
• Swapping the positions of the base and the argument, which calculates the logarithm with the argument as the base instead.
• Skipping verification by substitution, so an incorrect exponent goes unnoticed.

Related

n-math-log-value