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 + 2} = 434
the solution set expressed in terms of logarithms is
(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.)

Explanation:

Step1: Take log on both sides

Take the natural logarithm (ln) of both sides of the equation \(5^{x + 2}=434\). So we get \(\ln(5^{x + 2})=\ln(434)\).

Step2: Apply logarithm power rule

Using the power rule of logarithms \(\ln(a^b)=b\ln(a)\), the left - hand side becomes \((x + 2)\ln(5)=\ln(434)\).

Step3: Solve for x

First, divide both sides by \(\ln(5)\): \(x + 2=\frac{\ln(434)}{\ln(5)}\). Then, subtract 2 from both sides: \(x=\frac{\ln(434)}{\ln(5)}-2\).

Answer:

\(x=\frac{\ln(434)}{\ln(5)}-2\) (or if using common logarithms, \(x = \frac{\log(434)}{\log(5)}-2\))