Problem 50 - 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. $$\log _{5}(x-7)=2$$

Step-by-Step Solution

Answer

The exact solution for \(x\) is \(32\). The decimal approximation will also be \(32\) to two decimal places, provided it is within the domain of the original logarithmic equation.

Step 1: Convert the Logarithmic Equation to Exponential Form

To solve for \(x\), the equation, \(\log _{5}(x-7)=2\), is converted into its equivalent exponential form. This is done using the principle \(b^{y}=x \) is equivalent to \(\log_{b} x = y\). Hence, the following equivalent expression is obtained: \(5^2 = x - 7\).

Step 2: Solve for \(x\)

Proceeding with the obtained expression from Step 1, \(5^2 = x - 7\), we simplify this by calculating the power of 5, and then add 7 to both sides to isolate \(x\), which gives us the equation: \(x = 5^2 + 7\).

Step 3: Calculate the Value of \(x\)

Substituting 5 squared with 25, the equation becomes: \(x = 25 + 7\). Solve this to obtain \(x\).

Step 4: Check Validity of the Solution

With the obtained value of \(x\), substitute it back into the original logarithmic equation to verify if it satisfies the equation and that it is within the domain of the original logarithmic expression. A quantity is in the domain of a logarithmic function \(log_b x\) only if \(x\) is positive, therefore we need to confirm that \(x - 7 > 0\). In other words, \(x\) should be more than 7.

Step 5: Decimal Approximation

Since no additional simplification is required for this equation, the decimal approximation to two places will be the same as the exact answer derived from Step 3.