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.
\[
7^{x}=109
\]
Answer
\(\boxed{x \approx 2.41}\) is the solution to the equation \(7^{x}=109\).
Step 1 ：Given the exponential equation \(7^{x}=109\).
Step 2 ：We can use the property of logarithms that says the logarithm of a number to a certain base is the exponent to which the base must be raised to get the number. In other words, if \(b^y = x\), then \(\log_b{x} = y\). So, we can take the natural logarithm (or common logarithm) of both sides of the equation to solve for \(x\).
Step 3 ：Applying this to our equation, we get \(x = \log_{7}{109}\).
Step 4 ：Using a calculator to solve this, we get \(x \approx 2.41\).
Step 5 ：\(\boxed{x \approx 2.41}\) is the solution to the equation \(7^{x}=109\).
