Problem 53 - 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. $$\log _{4}(x+5)=3$$

Step-by-Step Solution

Answer

The exact solution to the logarithmic equation \(\log _{4}(x+5)=3\) is \(x=59\).

Step 1: Transform the equation from logarithmic to exponential form

We know from the properties of logarithms that \(\log _{b}a=c\) is equivalent to \(b^c=a\). So, we can write the given equation as \(4^3=x+5\). Now the equation is in a form we can solve for \(x\).

Step 2: Solve the equation for \(x\)

Evaluating \(4^3\), we get 64. So, the equation becomes \(64=x+5\). By isolating \(x\) on one side of the equation, the solution becomes \(x=64-5\). Hence, \(x=59\).

Step 3: Check the value of \(x\) to ensure it's in the domain of the original expression

The original expression is \(\log _{4}(x+5)\) which implies for any value of \(x, x+5 >0\). Plugging in our value of \(x=59\) into the equation, we get \(59+5=64\), which is greater than 0. Therefore, \(x=59\) is in the domain of the original logarithmic expressions.