Problem 21 - Precalculus

Question

Solve the equation analytically. $$ \log _{3}(x)=\log _{\frac{1}{3}}(x)+8 $$

Step-by-Step Solution

Answer

The solution is \( x = 81 \).

Step 1: Understand the Logarithmic Equation

The equation given involves two logarithms with different bases. The goal is to find the value of \( x \) that satisfies the equation \( \log_{3}(x) = \log_{\frac{1}{3}}(x) + 8 \).

Step 2: Substitute Logarithmic Property

Using the property of logarithms, \( \log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)} \), we can express \( \log_{\frac{1}{3}}(x) \) in terms of base 3:\[ \log_{\frac{1}{3}}(x) = \frac{\log_{3}(x)}{\log_{3}(\frac{1}{3})} = -\log_{3}(x), \] since \( \log_{3}\left(\frac{1}{3}\right) = -1 \).

Step 3: Simplify the Equation

Substitute the expression found in Step 2 into the equation:\[ \log_{3}(x) = -\log_{3}(x) + 8. \]

Step 4: Solve for \( \log_{3}(x) \)

Add \( \log_{3}(x) \) to both sides to isolate the logarithmic term: \[ 2\log_{3}(x) = 8. \]

Step 5: Divide Both Sides

Divide both sides of the equation by 2 to solve for \( \log_{3}(x) \):\[ \log_{3}(x) = 4. \]

Step 6: Convert to Exponential Form

Express \( \log_{3}(x) = 4 \) in exponential form: \[ x = 3^4. \]

Step 7: Calculate the Value of \( x \)

Calculate \( 3^4 \):\[ 3^4 = 81. \]

Step 8: Final Step: Verify the Solution

Substitute \( x = 81 \) back into the original equation to verify: - \( \log_{3}(81) = 4 \), because \( 3^4 = 81 \). - \( \log_{\frac{1}{3}}(81) = -4 \) because we found earlier that \( \log_{\frac{1}{3}}(x) = -\log_{3}(x) \). The equation \( 4 = -4 + 8 \) holds true.

Key Concepts

Properties of Logarithms

Logarithms have unique properties that simplify complex equations. They transform multiplication into addition, division into subtraction, and exponentiation into multiplication.

The two crucial properties used in this exercise are:

• Change of Base Property: This property allows you to rewrite a logarithm with a different base. For example, \[\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}\]You can pick any base \( c \), often 10 or \( e \), to perform these calculations.
• Negative Base Logarithm: The logarithm base that is the reciprocal of another results in a negative. For instance, \[\log_{3}\left(\frac{1}{3}\right) = -1\]because raising 3 to the power of \(-1\) gives \( \frac{1}{3} \).

These properties help rewrite the equation using the same base, which simplifies solving for unknowns.

Change of Base Formula

The change of base formula is crucial when dealing with logarithms of different bases. It helps you convert any logarithm to a more workable form.

In this exercise, we switch base \( \frac{1}{3} \) to base 3 because handling calculations in the same base simplifies the equation. The change is as follows:

• Using the property: \[\log_{\frac{1}{3}}(x) = \frac{\log_{3}(x)}{\log_{3}(\frac{1}{3})}\]
• Because \(\log_{3}\left(\frac{1}{3}\right) = -1\), this simplifies to \(-\log_{3}(x)\).

This transformation allows the combination of logarithmic expressions, ensuring the entire equation can be expressed uniformly in terms of \(\log_{3}(x)\). This is an important step towards simplifying and solving the equation.

Exponential Form

Switching from logarithmic to exponential form is essential for finding actual numeric solutions. Once you isolate the logarithmic term, converting it gives the result directly.

In our exercise, after simplifying:

• We reach \(\log_{3}(x) = 4\).
• Converting this to exponential form entails interpreting it as \(3^4\).
• This transition means "x is the number which 3 must be raised to equal 3 raised to the fourth power," so \[x = 81\].

Understanding this conversion is vital because it translates the logarithmic equation into a simpler arithmetic problem. This is often the concluding step in determining a number from a logarithmic equation.

Solve for x

Solving for \( x \) involves navigating through transformed equations and seeing them through to a specific value. In this exercise, applying logarithmic properties and conversions leads to an explicit solution.

Here's how it works step by step:

• Start with the transformed and simplified equation \[2\log_{3}(x) = 8\].
• Divide both sides by 2 to get \[\log_{3}(x) = 4\].
• Translate the logarithmic expression into its exponential form. Thus, \( x \) becomes \[ 3^4 = 81 \].
• Finally, verify by substituting back into the original equation, ensuring both sides match.

By understanding each transformation and keeping an eye on the bigger picture, you can solve \( x \) in such logarithmic equations easily.