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Chapter 11: Problem 14

Solve each equation. $$\log _{3}(x-4)+\log _{3}(x+4)=2$$

Short Answer

Expert verified
The solution is x = 5.

Step by step solution

01

Combine Logarithms

Use the property of logarithms \( \log_b(A) + \log_b(B) = \log_b(A \cdot B) \) to combine the logarithms on the left side of the equation:\[ \log_3((x-4)(x+4)) = 2 \]
02

Simplify Inside the Logarithm

Simplify the expression inside the logarithm, recognizing that \( (x-4)(x+4) \) is the difference of squares:\[ \log_3(x^2 - 16) = 2 \]
03

Exponentiate Both Sides

To remove the logarithm, exponentiate both sides with base 3:\[ x^2 - 16 = 3^2 \]
04

Solve for x

Solve for \(x \) by isolating \(x \):\[ x^2 - 16 = 9 \]\[ x^2 = 25 \]\[ x = 5 \text{ or } x = -5 \]
05

Verify Solutions

Substitute \(x = 5 \) and \(x = -5 \) back into the original equation to verify if they are valid solutions.For \( x = 5 \):\[ \log_3(5-4) + \log_3(5+4) = \log_3(1) + \log_3(9) = 0 + 2 = 2 \]For \( x = -5 \):\[ \log_3(-5-4) + \log_3(-5+4) = \log_3(-9) + \log_3(-1) \text{ (undefined as logs of negative numbers do not exist)} \]Thus, \( x = -5 \) is not a valid solution.

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Key Concepts

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

logarithmic properties
Logarithmic properties play a huge role in solving equations involving logs. The main properties you should know include:

  • Product Property: This property states that \[ \log_b(A) + \log_b(B) = \log_b(A \cdot B) \]. It allows you to combine the logs by multiplying the values inside them.
  • Quotient Property: This property shows that \[ \log_b(A) - \log_b(B) = \log_b(A/B) \]. It helps in dividing the values inside the logs when they are subtracted.
  • Power Property: It tells us that \[ \log_b(A^C) = C \cdot \log_b(A) \]. It allows us to move the exponent outside as a multiplier.

In our problem, we used the product property to combine two logarithms into one. This made it easier to simplify the equation, taking it from \( \log_3(x-4) + \log_3(x+4) = 2 \) to \( \log_3((x-4)(x+4)) = 2 \). Understanding these properties will make solving similar logarithmic equations more manageable and save you time!
difference of squares
The difference of squares is a specific polynomial form that simplifies the multiplication of binomials. The general formula is:

  • \[ (\ a - b \cdot a + b \ ) = a^2 - b^2 \]

In our problem, when combining the logs using the product property, we got an expression inside the logarithm: \( ( x-4 ) ( x +4 ) \). By recognizing this as a product of a binomial and its conjugate, we saw that:

\[ ( x-4)( x+4) = x^2 - 16 \]

This step is crucial for simplifying the logarithmic equation. By reducing the equation to a simpler polynomial, we made it easier to solve.
exponentiation
Exponentiation is the process of raising a number to a power. In the context of solving logarithmic equations, this process often helps to 'undo' a logarithm, simplifying the equation.

Here, once we simplified inside the logarithm and reached \( \log_3( x^2 -16 ) = 2 \), the next step was to use exponentiation to remove the log. By raising both sides to the power of 3, our equation transformed into:

\[ x^2 -16 = 3^2 \]

or

\[ x^2 -16 = 9 \]

This transform made solving for \( x \) straightforward, bringing us to:

\[ x^2 = 25 \]

and

\[ x = 5 \text{ or } x = -5 \]. Remember, exponentiation is very useful when you want to 'eliminate' a logarithm and simplify the equation further.

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Most popular questions from this chapter

Solve each problem. In a certain country the number of people above the poverty level is currently 28 million and growing \(5 \%\) annually. Assuming the population is growing continuously, the population \(P\) (in millions), \(t\) years from now, is determined by the formula \(P=28 e^{0.05 t},\) In how many years will there be 40 million people above the poverty level?

Determine whether each equation is true or false. $$ \log _{2}\left(16^{5}\right)=20 $$

Solve each equation. $$e^{x+3}=2$$

Find the exact solution and approximate solution to each equation. Round approximate answers to three decimal places. $$8^{x}=9^{x-1}$$

For each equation, find the exact solution and an approximate solution when appropriate. Round approximate answers to three decimal places. See the Strategy for Solving Exponential and Logarithmic $$\log (x-4)-\log (x+5)=1$$
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