Problem 74 - 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. $$3 \log _{2}(x-1)=5-\log _{2} 4$$

Step-by-Step Solution

Answer

The exact solution to the logarithmic equation is \(x = 3\).

Step 1: Combine logarithmic expressions

Firstly, combine the logarithms on both sides of the equation to simplify it. Remember to use the correct properties of logarithms - by applying the rules of logarithms: \(\log_a{mn} = \log_a{m} + \log_a{n}\) and \(\log_a{m^n} = n\log_a{m}\) we obtain: \(3 \log _{2}(x-1)+\log _{2} 4=5\). By noting that \(\log _{2} 4 = 2\), we simplify the equation to: \(3 \log _{2}(x-1)+2=5\)

Step 2: Isolate the logarithm

Next, isolate the logarithmic expression on one side of the equation. We do this by subtracting 2 from both sides of the equation and then dividing the entire equation by 3: \(\log _{2}(x-1) = 1\).

Step 3: Remove the logarithm

To remove the logarithm, we can use exponentiation, by raising 2 (the base of the logarithm) to the power of both sides: \(2^{\log _{2}(x-1)} = 2^{1}\), which simplifies to \(x - 1 = 2\).

Step 4: Solve for \(x\)

Finally, we solve for \(x\) by adding 1 to both sides of the equation. This provides the result: \(x = 3\)

Step 5: Check the solution

It is crucial to ensure that the solution is within the domain of the original logarithmic expression. Observing that plugging \(x = 3\) into the original expression (\(x - 1 = 3 - 1 = 2\)), which indeed is in the domain of a logarithmic function - ensuring that the solution \(x = 3\) is valid.