Problem 41 - 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. $$\log _{3} x=4$$

Step-by-Step Solution

Answer

The exact value solution for the equation \(\log_{3} x = 4\) is \(x = 81\).

Step 1: Understand the properties of the logarithm in the equation

The equation is in logarithmic form with base 3. Logarithms and exponentials are inverses of each other, meaning that they can be rewritten in terms of each other. So, the equation can be written in exponential form as \(3^4 = x\).

Step 2: Calculate the exponential expression

Calculate \(3^4\), which involves raising 3 to the power of 4. This gives \(3*3*3*3 = 81\).

Step 3: Check the domain

Since the input of a logarithm (x in this case) should be greater than zero, the domain of the logarithmic function is \(x > 0\). In this case, since 81 is greater than 0, the solution is valid.