Problem 81 - 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. $$2 \log x=\log 25$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation is \(x = 5\).

Step 1: Simplify the Equation

Utilizing log properties, rewrite the equation as \(log(x^2) = log(25)\).

Step 2: Equating Inside Arguments

By the property of equality of logarithms, if two logs have the same base and are equal, then their arguments are equal. So, equating the inside arguments we get \(x^2 = 25\).

Step 3: Solve for x

Taking the square root on both sides gives \(x = \pm5\). However, since \(x\) represents the argument of a logarithm and this value must always be positive, we discard the negative solution. Hence, \(x = 5\) is the solution.

Step 4: Check the Solution with a Calculator

If necessary, a calculator can be used to confirm the solution. The value of \(2 \log (5)\) should be the value of \(\log (25)\), i.e., both should give the same decimal value when computed in a calculator.