Problem 49 - 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. $$\log _{4}(x+5)=3$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 59\).

Step 1: Convert Logarithmic Form to Exponential Form

By converting logarithmic form to exponential form, we have \(4^{3} = x+5\). This makes it easier to express and solve the equation.

Step 2: Calculate the Real Value

Following from the step above, calculate \(4^{3} = 64\). Hence we have \(64 = x+5\).

Step 3: Solve for x

On rearranging the equation obtained in step 2 to get the value of x, we subtract 5 from both sides and we get \(x = 64 - 5 = 59\).

Step 4: Confirm if x is in original Domain

The original domain for the logarithmic function would be \(x > -5\), since within the bracket, x+5, the values must be greater than zero. The solution found \(x = 59\) is in the domain of original logarithmic expressions as it is greater than -5.