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Chapter 4: Problem 39

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$7^{0.3 x}=813$$

Short Answer

Expert verified
The solution for \(x\) in terms of natural logarithms is \(x = \frac{\ln(813)}{0.3 \ln(7)}\). As for the decimal approximation of \(x\), it must be calculated using a calculator.

Step by step solution

01

Apply the natural logarithm

Use logarithmic form to write the given equation. It becomes: \(\ln(7^{0.3x}) = \ln(813)\).
02

Use the properties of logarithms

Apply the power rule of logarithms to bring the exponent out front: \(0.3x \times \ln(7) = \ln(813)\).
03

Solve for x

Isolate \(x\) by dividing both sides of the equation by \(0.3 \ln(7)\). This results in \(x = \frac{\ln(813)}{0.3\ln(7)}\).
04

Calculate decimal approximation

Use a calculator to determine the decimal approximation of \(x\).

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Most popular questions from this chapter

Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$

Would you prefer that your salary be modeled exponentially or logarithmically? Explain your answer.

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I estimate that \(\log _{8} 16\) lies between 1 and 2 because \(8^{1}=8\) and \(8^{2}=64\).

Evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$\log _{3} 81, \text { or } \log _{3} 9^{2} ?$$
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