Problem 38 - College Algebra Essentials

Question

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5^{x-3}=137$$

Step-by-Step Solution

Answer

The solution to the exponential equation is approximately \( x = 5.61 \).

Step 1: Apply the natural logarithm to both sides of the equation

Taking the natural logarithm on both sides of the equation will help in getting the variable out of the exponent. Using the law of logarithms, \( ln(a^b)=b*ln(a) \), the equation will transform from \(5^{x-3}=137\) into \( (x-3) * ln(5) = ln(137) \).

Step 2: Isolate the variable

To isolate x, the equation is divided by \( ln(5) \) on both sides and add 3. This changes the equation to \( x = ln(137) / ln(5) + 3 \)

Step 3: Calculate the solution using a calculator

Use a calculator to compute the value of \( x \), keeping the answer to two decimal places. This gives us the decimal approximation of the solution.