Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Give the exact answer. $$\log _{9} \frac{\sqrt[4]{27}}{3}=x$$

Step-by-Step Solution

Answer

\( x = -\frac{1}{8} \)

Step 1: Simplify the Fraction Inside the Logarithm

We start by simplifying the expression inside the logarithm: \( \frac{\sqrt[4]{27}}{3} \). First, express \( \sqrt[4]{27} \) as a power of 3. We can rewrite 27 as \( 3^3 \), so \( \sqrt[4]{27} = (3^3)^{1/4} = 3^{3/4} \). Now, the entire fraction becomes \( \frac{3^{3/4}}{3} \). Simplify it to get \( 3^{3/4 - 1} = 3^{-1/4} \).

Step 2: Apply the Logarithmic Identity

The equation now looks like \( \log_{9}(3^{-1/4}) = x \). Recall that a logarithmic identity states \( \log_b(a^c) = c \cdot \log_b(a) \). Applying this identity gives us \(-\frac{1}{4} \cdot \log_{9}(3) = x\).

Step 3: Change the Base of the Logarithm

To solve \( \log_{9}(3) \), rewrite it using the base change formula: \( \log_{9}(3) = \frac{\log_{3}(3)}{\log_{3}(9)} \). Since \( \log_{3}(3) = 1 \) and \( 9 = 3^2 \), we have \( \log_{3}(9) = \log_{3}(3^2) = 2 \cdot \log_{3}(3) = 2 \). Thus, \( \log_{9}(3) = \frac{1}{2} \).

Step 4: Substitute Back to Solve for x

Substitute back the value of \( \log_{9}(3) \) into \( -\frac{1}{4} \cdot \log_{9}(3) = x \). This yields \( x = -\frac{1}{4} \cdot \frac{1}{2} = -\frac{1}{8} \).

Key Concepts

Logarithmic Identities

Logarithmic identities are powerful tools for simplifying and solving logarithmic equations. One key identity used in the provided solution is:

• \( \log_b(a^c) = c \cdot \log_b(a) \)

This identity tells us that we can bring the exponent down in front of the logarithm as a multiplier. In simpler terms, if you have a power inside a log, like \( a^c \), you can move the exponent \( c \) in front of the log. This greatly simplifies complex expressions.
Here's how this looks in practice: if you have \( \log_{9}(3^{-1/4}) \), this can be rewritten as \( -\frac{1}{4} \times \log_{9}(3) \). The negative sign indicates that the exponent was originally negative.
By understanding this identity, solving logarithmic equations becomes more straightforward, since it allows you to work directly with familiar numbers instead of trying to solve an abstract equation.

Base Change Formula

The base change formula is another essential concept when working with logarithms. It allows you to convert logarithms from one base to another, which is particularly useful when you need to evaluate a logarithm but the base isn't easy to work with. The formula is:

• \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \)

Here, \( k \) can be any positive number, but typically \( 10 \) or \( e \) (the natural logarithm base) are used for simplicity due to calculator capabilities.
In the solution, \( \log_{9}(3) \) was changed to base 3 using this formula:

• \( \log_{9}(3) = \frac{\log_{3}(3)}{\log_{3}(9)} \)

This conversion is critical because \( \log_{3}(3) \) simplifies to \( 1 \), and \( \log_{3}(9) \) simplifies to \( 2 \) (since \( 9 = 3^2 \)). This further simplifies \( \log_{9}(3) \) into \( \frac{1}{2} \). Using base change can thus make calculations much simpler and clearer.

Exponents and Radicals

Understanding exponents and radicals is key when handling logarithmic functions, as they often appear together. Exponents represent power and are used to denote repeated multiplication, while radicals are used to indicate roots. Understanding how to convert between them is crucial:

• \( a^{1/n} = \sqrt[n]{a} \)

This allows us to turn roots into fractional exponents, simplifying expressions for logarithmic operations. For example, in the given problem, \( \sqrt[4]{27} \) is converted to an exponential form: \( (3^3)^{1/4} = 3^{3/4} \).
This conversion allowed further simplification of the expression inside the logarithm.
By understanding how to manipulate and convert between exponents and radicals, you simplify calculations and make it easier to apply logarithmic identities and other algebraic techniques.