Problem 27 - 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. $$5^{x}=17$$

Step-by-Step Solution

Answer

The solution to the equation \(5^x = 17\) is x = \( \ln(17) / \ln(5)\). Using a calculator, we get: x ≈ 1.76.

Step 1: Convert the exponential equation into logarithmic form

An exponential equation can always be written as a logarithmic equation and vice versa. This is because they are inverse operations. So, \(5^{x} = 17 \) can be rewritten using a logarithm base 5: \( \log_{5}(17) = x \)

Step 2: Convert the Base of the Logarithm to Use Natural Logarithm

The change of base formula says that \( \log_{b}(a) = \log_{k}(a) / \log_{k}(b) \). We can use natural logarithms (base e) here to change the base since that's what's easily computed on most calculators : \( \log_{5}(17) = \ln(17) / \ln(5) = x\)

Step 3: Use a Calculator to Obtain Decimal Value

Once you have obtained the natural logarithms, you can use a calculator to compute the value of x. Be sure to round your answer to two decimal places as per the problem's instructions.