Problem 58 - 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. $$\log _{7}(x+2)=-2$$

Step-by-Step Solution

Answer

The logarithmic equation \(\log _{7}(x+2)=-2\) has no solution because any obtained solution doesn't comply with the constraints of the logarithmic domain.

Step 1: Rewrite the logarithm into equivalent exponential form

From the basic logarithm form, \( log_{b}a = c \) is equivalent to \(b^c = a\). So using this property we can translate our initial problem \( \log _{7}(x+2)=-2 \) into exponential form, giving \(7^{-2} = x + 2\).

Step 2: Simplify the equation

Now, calculate \(7^{-2}\) which is \( \frac{1}{7^2} \) or \( \frac{1}{49} \). This makes our equation become \( \frac{1}{49} = x + 2 \).

Step 3: Solve for \(x\)

In this step, isolate \(x\) by subtracting 2 from both sides of the equation. This results in: \( x = \frac{1}{49} - 2 \).

Step 4: Rearrange the equation for easy computation

To effectively subtract the 2, a common denominator is required. Which means rewriting the 2 as \( \frac{98}{49} \). This yields the new equation: \( x = \frac{1}{49} - \frac{98}{49} \).

Step 5: Calculate the value of \(x\)

Carry out the subtraction gives the answer: \( x = -\frac{97}{49} \).

Step 6: Check the domain

The domain of the original logarithmic equation is \( x + 2 > 0 \). If we substitute \( x = -\frac{97}{49} \) we see that it is not in the domain because \( -\frac{97}{49} + 2 = -\frac{49}{49} = -1 \) which is not greater than zero. So, this value is rejected.