Problem 78 - 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. $$\log (5 x+1)=\log (2 x+3)+\log 2$$

Step-by-Step Solution

Answer

The solution for the logarithmic equation is \(x = 5\)

Step 1: Combine Logarithms on the right side of the equation

The first step is applying the properties of logarithms to combine the two logarithms on the right side by using the property \(\log(a)+\log(b) = \log(ab)\). So, the equation becomes: \[\log (5x + 1) = \log(2(2x + 3))\]

Step 2: Remove the logarithms

Since the logarithms on each side of the equation have the same base, we can use the property that if \(\log_ba = \log_bc\), then \(a = c\) to remove the logarithms. Therefore, we have: \[5x + 1 = 2(2x + 3)\]

Step 3: Solve the equation for \(x\)

Solving this equation gives the following steps: start by expanding the right side, which gives:\[5x + 1 = 4x + 6\]Subtract \(4x\) from both sides of the equation:\[x + 1 = 6\]Finally, subtract 1 from both sides, giving:\[x = 5\]

Step 4: Check the solution

To be sure that the solution is correct, check that \(x=5\) is in the domain of the original logarithmic expressions by substituting \(x=5\) into the original equation and verifying that it gives a true statement.