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Chapter 3: Problem 23

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$3^{2 x}=80$$

Short Answer

Expert verified
When evaluated, \(x\) approximates to 3.184

Step by step solution

01

Write the equation

We first write down the given equation which is: \(3^{2x} = 80\)
02

Transform the equation to logarithmic form

Transformation from exponential to logarithmic form will allow us to begin isolating x. The conversion equation from the exponential form \(b^{y} = x\) to the logarithmic form is \(\log_b(x) = y\). Using this, our transformed equation is: \(\log_3(80) = 2x\)
03

Isolate x

In order to isolate x, we divide both sides of the equation by 2. This yields: \(x = \frac{\log_3(80)}{2}\)
04

Evaluate using a calculator

Lastly, using a calculator to evaluate the expression above, don't forget that the result should be approximated to three decimal places.

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Most popular questions from this chapter

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{-0.9}=0.406 \ldots$

Write the logarithmic equation in exponential form. $$\ln 250=5.521 \ldots$$

Determine whether the statement is true or false. Justify your answer. The graph of a Gaussian model will never have an \(x\) -intercept.

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$g(x)=\ln (-x)$$

Using the One-to-One Property In Exercises \(73-76,\) use the One-to-One Property to solve the equation for \(x\). .\(r\)\ln \left(x^{2}-2\right)=\ln 23$
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