Problem 81 - Introductory and Intermediate Algebra for College Students 4th

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 valid solution for the given logarithmic equation is \(x = 5\).

Step 1: Simplify the Equation

A property of logarithms states that \(\log_b (a^n) = n \log_b(a)\). By utilizing this, the given equation \(2 \log x=\log 25\) can be rewritten as: \(\log (x^2) = \log 25\).

Step 2: Isolate x

Since log function is a one-to-one function, if \(\log (x^2) = \log 25\), then it must be that \(x^2 = 25\).\nTo solve for \(x\), we consider the possible square roots of 25, which are \(-5\) and \(5\).

Step 3: Eliminate Extraneous Solutions

In the domain of logarithmic expressions, the argument of the logarithm must always be greater than 0. Therefore, \(x = -5\) is not a valid solution because it would make the argument of the original logarithmic expressions negative. So we reject the solution \(x = -5\).

Step 4: Express Solution in Decimal

The remaining solution \(x = 5\) does not need to be expressed in decimal form as it is an exact integer value.