Problem 50 - College Algebra Essentials

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{5} x=3$$

Step-by-Step Solution

Answer

The exact solution to the logarithmic equation is \(x = 125\).

Step 1: Write in Exponential Form

The given equation is \(\log _{5} x = 3\). Remember that logarithmic form and exponential form are equivalent. So, this equation can be rewritten in exponential form as \(5^3 = x\).

Step 2: Solve for x

Using the calculation \(5^3 = 125\), we can solve for \(x\), which results in \(x = 125\).

Step 3: Check the Solution

Since the value for \(x\) is a positive number, it doesn't need to be rejected as it falls in the domain of the original logarithmic expressions. So, \(x = 125\) is the solution to the logarithmic equation.