Problem 84 - 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. $$\log (x+7)-\log 3=\log (7 x+1)$$

Step-by-Step Solution

Answer

The exact solution to the given logarithmic equation is \(x = 1/4\) or as a decimal, \(x = 0.25\)

Step 1: Apply the property of logarithms

Using the property of logarithms, \(\log a - \log b = \log (a/b)\), the equation can be rewritten as \(\log ((x+7)/3) = \log (7x+1)\)

Step 2: Equate the arguments of the logarithms

Since the logarithms on both sides of the equation are equal, this implies their arguments are equal. Therefore, we get the equation \((x+7)/3 = 7x+1\)

Step 3: Solve for x

To solve for \(x\), first multiply both sides of the equation by 3 to get rid of the denominator on the left side, leading to \(x+7 = 21x + 3\). After rearranging, we obtain \(x = 1/4\)

Step 4: Validation of the solution

We need to make sure \(x=1/4\) satisfies the original logarithmic equation. Checking for \(x=1/4\) in the original equation, we see that each logarithm has a positive argument, which means \(x=1/4\) is a valid solution and does not need to be rejected