Evaluate each expression without using a calculator. log₃ 27

Verified step by step guidance

1. Recognize that the expression is \( \log_{3} 27 \), which asks: "To what power must 3 be raised to get 27?"
2. Rewrite 27 as a power of 3. Since \( 27 = 3^3 \), substitute this into the logarithm: \( \log_{3} 3^3 \).
3. Use the logarithmic identity \( \log_{a} a^x = x \) to simplify the expression.
4. Apply the identity to get \( \log_{3} 3^3 = 3 \).
5. Conclude that the value of \( \log_{3} 27 \) is the exponent 3, since \( 3^3 = 27 \).

Key Concepts

Logarithm Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log base 3 of 27 asks, '3 raised to what power equals 27?' Understanding this definition is essential to evaluate logarithmic expressions.

Exponentiation and Powers

Recognizing powers of numbers helps simplify logarithms. Since 27 is 3 raised to the 3rd power (3^3 = 27), this relationship allows direct evaluation of log base 3 of 27 by identifying the exponent.

Properties of Logarithms

Logarithms have properties such as log_b(b^x) = x, which means the logarithm of a base raised to an exponent is just that exponent. This property simplifies evaluating expressions like log_3(27) by converting the problem into finding the exponent.