Problem 82 - 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. $$3 \log x=\log 125$$

Step-by-Step Solution

Answer

The exact solution for the logarithmic equation \( 3 \log x = \log 125 \) is \( x = 5 \).

Step 1: Simplify the equation

We note that both sides of the equation are equal to logarithms, this allows us to simplify the equation. We can use the power rule of logarithms to move the 3 from the left side to the power of \(x\). This gives us a new equation that is \( \log (x^3) = \log (125) \).

Step 2: Remove logarithms

With the equation in this form and since the logarithms on both sides of the equation have the same base, we can remove the logarithms \( \log \) from both sides of the equation. This leaves us with \( x^3 = 125 \).

Step 3: Solve for x

To solve for \(x\), we should take the cube root of both sides of the equation. This gives \( x = \sqrt[3]{125} \). It's advisable to use a calculator to get the exact value of the root. The cube root of \(125\) gives \(x = 5\).

Step 4: Verification

We need to substitute the calculated value back into the original equation to confirm if it is a valid solution. Substituting \(x = 5\) into the original equation; \( 3 \log x = \log 125 \) gives \( 3 \log 5 = \log 125 \). Since \( \log 125 = 3 \log 5 \), the answer is verified. In addition, \(5\) is within the domain of the original logarithmic expression.