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Chapter 4: Problem 95

Solve each equation. Find the exact solutions. $$3^{x}=77$$

Short Answer

Expert verified
\(x = \frac{\ln(77)}{\ln(3)}\)

Step by step solution

01

Take the natural logarithm of both sides

To solve the equation, we first take the natural logarithm of both sides to utilize the properties of logarithms. Thus, \(\ln(3^x) = \ln(77)\).
02

Apply logarithm properties

Use the power rule of logarithms, \(\ln(a^b) = b \ln(a)\), to simplify the equation: \(x \ln(3) = \ln(77)\).
03

Solve for x

Finally, isolate \(x\) by dividing both sides of the equation by \(\ln(3)\): \(x = \frac{\ln(77)}{\ln(3)}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
In the given exercise, we use the natural logarithm to solve an exponential equation.

The natural logarithm, written as \(\text{ln}(x)\), is a logarithm to the base \(\text{e}\), where \(\text{e}\) is an irrational constant approximately equal to 2.71828. It helps to solve equations where the variable is an exponent.

Taking the natural logarithm of both sides of an equation like \(3^{x}=77\) helps us apply logarithmic properties to linearize the equation.
Logarithm Properties
Logarithms have several key properties that make them useful for solving equations involving exponents. For this equation, \(3^{x}=77\), we use:
  • Power Rule: This states that \(\text{ln}(a^b) = b \text{ln}(a)\). We apply this rule to transform the exponential form into a linear form. So \(\text{ln}(3^x)\) becomes \(x \text{ln}(3)\).

    Other useful properties to know include:
  • Product Rule: \(\text{ln}(ab) = \text{ln}(a) + \text{ln}(b)\).
  • Quotient Rule: \(\text{ln}\frac{a}{b} = \text{ln}(a) - \text{ln}(b)\).
By knowing and applying these properties, solving exponential equations becomes much simpler.
Power Rule
The Power Rule of logarithms is utilized extensively in solving exponential functions.

In the given problem: After taking the natural logarithm of both sides \( (\text{ln}(3^x) = \text{ln}(77))\), we use the power rule which transforms \(x \text{ln}(3) = \text{ln}(77)\).

Steps to use Power Rule:
  • Identify the term with an exponent.
  • Apply the natural logarithm to both sides of the equation.
  • Use the Power Rule \( \text{ln}(a^b) = b \text{ln}(a) \) to simplify.
  • Isolate the variable by performing necessary operations on both sides.


The final step involves isolating \(x\) by dividing both sides by \( \text{ln}(3)\) to get \( x = \frac{\text{ln}(77)}{\text{ln}(3)}\). This gives us the solution to the original exponential equation.

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Most popular questions from this chapter

Work in a small group to write a summary (including drawings) of the types of graphs that can be obtained for logarithmic functions of the form \(y=\log _{a}(x)\) for \(a>0\) and \(a \neq 1\).

Find the approximate solution to each equation. Round to four decimal places. $$\frac{1}{e^{x-1}}=5$$

Evaluate \(\left(2 \times 10^{-9}\right)^{3}\left(5 \times 10^{3}\right)^{2}\) without a calculator. Write the answer in scientific notation.

Depreciation and Inflation Boris won a \(\$ 35,000\) luxury car on Wheel of Fortune. He plans to keep it until he can trade it evenly for a new compact car that currently costs \(\$ 10,000\). If the value of the luxury car decreases by \(8 \%\) each year and the cost of the compact car increases by \(5 \%\) each year, then in how many years will he be able to make the trade?

Find the approximate rational solution to the equation \(1.56^{x-1}=9.8 .\) Round the answer to four decimal places.
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