Problem 42 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$7^{\log _{7} 23}$$

Step-by-Step Solution

Answer

The expression \(7^{\log _{7} 23}\) evaluates to 23.

Step 1: Understand the expression

In the expression \(7^{\log _{7} 23}\), the base of the exponent and the base of the logarithm are the same. The logarithm \(\log _{7} 23\) is simply asking 'to what power must we raise 7 to obtain 23?' And that is the power to which 7 is being raised when we evaluate \(7^{\log _{7} 23}\).

Step 2: Apply the Exponent and Logarithm Identity

The identity \(a^{\log _{a} x} = x\) tells us that any base 'a' raised to the power of its logarithm in the same base of what we denote as 'x', will always equals to 'x'. This is true in our case since the base of the exponent and the base of the logarithm are the same, which is 7; so \(7^{\log _{7} 23} = 23\).