Problem 73 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$\log _{3}(x-1)=2$$

Step-by-Step Solution

Answer

The equivalent exponential form of the equation is \(3^2 = x - 1\). Solving for x, we find that \(x = 10\).

Step 1: Convert Logarithmic to Exponential Form

To convert this equation from its logarithmic form \(\log_{3}(x-1)=2\) into its equivalent exponential form, apply the general rule of logarithms that \(\log_{b}(x)=y\) is equivalent to b^y=x. Thus, the equation in exponential form is \(3^2 = x - 1\).

Step 2: Evaluate the Exponential

Now that the equation is in exponential form, calculate \(3^2\) to obtain a numerical value. This results into \(9 = x - 1\).

Step 3: Solve for x

The final step is to isolate x in the equation \(9 = x - 1\). This can be done by adding 1 to both sides to solve for x, which gives \(x = 9 + 1\), therefore \(x = 10\).