Problem 49 - 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. $$\log _{3} x=4$$

Step-by-Step Solution

Answer

The exact solution is x = 81.

Step 1: Convert logarithmic equation to exponential form

To convert the logarithmic equation to its equivalent exponential form, use the conversion rule. In this case, the base is 3, the logarithm itself is x, and is equal to 4. Thus, the equivalent exponential form is \(3^4 = x\).

Step 2: Simplify to find the value of x

Solve the exponential equation to find the value of x, calculating 3 raised to the power of 4 which gives \(3^4 = 81\). Hence, x = 81.

Step 3: Check the solution

Substitute x = 81 back into the original logarithmic equation to verify the solution. \(\log_3 81 = 4\) since \(3^4 = 81\). So, the solution is valid.

Step 4: Decimal approximation

As the solution is an integer, there's no need to provide a decimal approximation here.