Problem 52 - 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. $$\log _{7}(x+2)=-2$$

Step-by-Step Solution

Answer

The solution to the equation \(\log _{7}(x+2)=-2\) is \(x=\frac{1}{49}-2\). After performing the validation step, it should be confirmed that the obtained solution is within the domain of the logarithmic expression, and that it correctly balances the initial equation.

Step 1: Convert Logarithmic Equation to Exponential Form

Logarithmic equations can be transformed to exponential equations using the definition of logarithms which states: if \(\log_b(c)=a\), then it is the same as \(b^a=c\). So, the given equation \(\log _{7}(x+2)=-2 \) becomes \(7^{-2}=x+2\).

Step 2: Simplify the Exponential Equation

The exponential equation \(7^{-2}=x+2\) can be simplified by calculating the value of \(7^{-2}\), which is equal to \(\frac{1}{49}\). This results in the equation \(\frac{1}{49}=x+2\).

Step 3: Solve for the Variable

To find \(x\), isolate this variable by subtracting 2 from both sides of the equation. You obtain: \(x = \frac{1}{49} - 2\). Calculate the value of this expression.

Step 4: Validate the Solution

Replace \(x\) in the initial logarithmic equation \(\log _{7}(x+2) =-2\) with the computed value. Compute to confirm that the value obtained is indeed -2. If it isn't, the original value of \(x\) is not valid within the domain of the logarithmic expression, in which case it will need to be disregarded.