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Chapter 12: Problem 73

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

Short Answer

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

Step by step solution

01

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 \(3^2 = x - 1\).
02

Evaluate the Exponential

Now that the equation is in exponential form, calculate \(3^2\) to obtain a numerical value. This results into \(9 = x - 1\).
03

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 \(x = 10\).

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Most popular questions from this chapter

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(3 x-2)=2$$

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$2^{x+1}=8$$

Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

What is a logarithmic equation?

$$\text { Solve: } x-2(3 x-2)>2 x-3$$
See all solutions

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