Problem 83 - 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. $$\log (x+4)-\log 2=\log (5 x+1)$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2/9\) or approximately \(x = 0.22\) when expressed to two decimal places.

Step 1: Simplifying the Equation

First, simplify the equation using the property of subtraction of logarithms. The equation \( \log (x+4)-\log 2 = \log (5x+1) \) can be rewritten as \( \log \frac{x+4}{2} = \log (5x+1) \)

Step 2: Solving the Equation

Once the logarithms on both sides have the same base, the equations inside the logarithms can be equated to one another. That gives the equation \( \frac{x+4}{2}=5x+1 \). Solving this equation, let's first multiply through by 2 to clear the fraction, resulting in \(x + 4 = 10x + 2\).

Step 3: Finding the Value of x

Continue solving the equation for \( x \). Subtract \( x \) from both sides of the equation, resulting in \( 4 = 9x + 2 \). Then subtract 2 from both sides to get \( 2 = 9x \). Finally, divide by 9 to find \( x = 2/9 \).

Step 4: Checking Solution in Original Equation

Make sure that this solution doesn't make the argument of the original logarithmic expressions undefined or negative. Here, \(x = 2/9\) is acceptable for all expressions.

Step 5: Calculating Decimal Approximation

This step may not be necessary for all problems, but when asked, calculate the decimal approximation. In this case, \(x = 2/9 ≈ 0.22\), to two decimal places.