Problem 54 - 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. $$\log _{5}(x-7)=2$$

Step-by-Step Solution

Answer

The exact solution for the logarithmic equation is \(x = 32\).

Step 1: Understand the Basis of Logarithms

A logarithm is an operation that attempts to find out the exponent for a given base to yield the argument. In essence, an equation in the form of \(\log _{b}(a)=c\) is equivalent to \(b^{c}=a\).

Step 2: Convert Logarithmic Equation Into Exponential Form

This equation, \(\log _{5}(x - 7) = 2\), can be written in exponential form as \(5^2 = x - 7\). Hence, we get \(25 = x - 7\).

Step 3: Solve for x

Add 7 to both sides of the equation to solve for x, \(x = 25 + 7\). Thus, \(x = 32\).

Step 4: Check for the Validity of the Solution

Substitute x=32 to the original equation \(\log _{5}(32 - 7)\). It simplifies to \(\log _{5}(25)\), which equals to 2. So, the solution is valid.

Step 5: Provide Exact and Approximate Answers

The exact solution is \(x = 32\). Since it doesn't involve irrational numbers, there's no need for a decimal approximation.