Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The evaluated expression is 23.

Step 1: Identify the Logarithmic Identity

One of the identities of logarithm is \(b^{\log_{b} a} = a\), where b is the base, and a is the index. The given expression can be compared to this identity. In this case, the base of the power (b) is 7 and the index (a) inside the logarithm is 23. It can therefore be seen that the base of the logarithm is the same as the base of the power, which confirms that we can indeed use this logarithmic identity to simplify the expression.

Step 2: Apply the Logarithmic Identity

Apply the logarithmic identity \(b^{\log_{b} a} = a\) to the given expression \(7^{\log _{7} 23}\). By this identity, the result is the number inside the logarithm, which is 23.