Question

solve the following exponential equation. express the solution set in terms of natural logarithms or common logarithms. then, use a calculator to obtain a decimal approximation for the solution.
$5^x = 79$

the solution set expressed in terms of logarithms is \\(\left\\{\frac{\log(79)}{\log(5)}\
ight\\}
(use a comma to separate answers as needed. simplify your answer. use integers or fractions for any numbers in the expression. use \\(\ln\\) for natural logarithm and \\(\log\\) for common logarithm.)
now use a calculator to obtain a decimal approximation for the solution.
the solution set is \\(\square\\)
(use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Recall the logarithm formula

We know that if \(a^x = b\), then \(x=\frac{\log(b)}{\log(a)}\) (using common logarithms) or \(x = \frac{\ln(b)}{\ln(a)}\) (using natural logarithms). Here we have the expression \(\frac{\log(79)}{\log(5)}\) from the problem.

Step2: Calculate the value

First, find \(\log(79)\) and \(\log(5)\) using a calculator. \(\log(79)\approx1.8976\) and \(\log(5)\approx0.6990\). Then divide these two values: \(\frac{1.8976}{0.6990}\approx2.71\) (rounded to two decimal places).

Answer:

\(2.71\)