Logarithm of 81 to base 3

What is the value of the logarithm base 3 of 81? Verify your answer by rewriting it in exponential form.

Step-by-step

1. Base a = 3, argument N = 81
   For logarithms, the base must be greater than zero and not equal to one, and the argument must be greater than zero.
2. log_3 81 = 4
   A logarithm is the inverse of exponentiation; it asks what power of the base equals the argument.
3. Check: 3^4 = 81
   Substitute the result back into exponential form to quickly catch calculation errors.
4. Domain check: base 3 is greater than zero and not equal to one, argument 81 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

A logarithm asks "to what power must the base be raised to equal the argument?" so it is fundamentally the inverse operation of exponentiation. The method is to rewrite the argument as a power of the base; that exponent is the logarithmic value. The base must be greater than zero and not equal to one, and the argument must be greater than zero. These are domain requirements, and if they are not met, the logarithm is undefined.

Common mistakes

• Confuse the logarithmic value with the argument itself, writing down eighty-one directly.
• Only write the result without back-substitution, so an incorrect exponent goes unnoticed.
• Ignore the restrictions on the base and argument values, and blindly compute for illegal inputs.

Related

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