Problem 55 - 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. $$\log _{2}(x+25)=4$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\log_{2}(x + 25) = 4\) is \(x = -9\).

Step 1: Convert to Exponential Form

We begin the process by converting the logarithmic equation \(\log _{2}(x+25)=4\) to its exponential form. This is done using the fact that \(b^y = x\) is equivalent to \(\log_b{x} = y\). Hence, \(2^4 = x + 25\).

Step 2: Solve for \(x\)

Solving the equation \(2^4 = x + 25\) for \(x\) provides the exact answer. Doing the math, we have \(16 = x + 25\), which simplifies down to \(x = -9\) when you subtract 25 from both sides.

Step 3: Verify the Domain

It's vital to check if our solution for \(x\) falls within the domain of the original logarithmic expression. Our logarithmic expression is \(\log_{2}(x+25)\). For any log base \(b\) of \(x\) (denoted as \(\log_b{x}\)), \(x > 0\). Therefore, \(x+25 > 0\). Solving for \(x\) gives \(x > -25\). Our solution, \(x = -9\), is greater than -25, so it falls within the domain.