Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each logarithm in three ways: (a) Use the definition of logarithm to find the exact value analytically. (b) Support the result of part (a) by using the change-of-base rule and common logarithms on a calculator. (c) Use a buill-in calculator finction to evaluate the logarithm. (d) Support the result of part (a) by locating the appropriate point on the graph of the finction \(y=\log _{a} x\). $$\log _{9} 27$$

Step-by-Step Solution

Answer

\( \log_{9}(27) = 1.5 \).

Step 1: Use the Definition of Logarithm

To find the exact value analytically, use the definition of a logarithm: \( \log_{a}(b) = c \) implies that \( a^c = b \). Here, we must find \( \log_{9}(27) \). Let this be \( x \). Thus, we have: \( 9^x = 27 \).Since \( 9 \) is \( 3^2 \) and \( 27 \) is \( 3^3 \), rewrite the equation:\( (3^2)^x = 3^3 \).This simplifies to \( 3^{2x} = 3^3 \). Since the bases are the same, equate the exponents:\( 2x = 3 \).Solve for \( x \):\( x = \frac{3}{2} \).Therefore, \( \log_{9}(27) = \frac{3}{2} \).

Step 2: Use Change-of-Base Formula

The change-of-base formula states: \( \log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)} \). Using common logarithms (base 10), the expression becomes:\[ \log_{9}(27) = \frac{\log_{10}(27)}{\log_{10}(9)} \].By calculating using a calculator:\( \log_{10}(27) \approx 1.431 \) and \( \log_{10}(9) \approx 0.954 \).Thus, \( \log_{9}(27) \approx \frac{1.431}{0.954} \approx 1.5 \), which confirms our result from Step 1.

Step 3: Use a Built-in Calculator Function

Most calculators have a function for calculating logarithms directly with a specific base:Enter \( \log_{9}(27) \) in the calculator with this function. This should directly provide the result: \( 1.5 \).

Step 4: Verify on the Graph

Graph the function \( y = \log_{9}(x) \).Identify the value of \( x = 27 \) and find the corresponding \( y \) value on the graph.At \( x = 27 \), \( y = 1.5 \).This supports the result \( \log_{9}(27) = 1.5 \) found in Steps 1 to 3.

Key Concepts

Change of Base Formula

Converting logarithms to a different base is made easy with the change of base formula. This formula is especially handy when using calculators that only compute logarithms in bases 10 or natural base e. The change of base formula states: \( \log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)} \). You can select any base \( c \) that suits your need, most commonly 10 or \( e \) for simplicity. This allows computational flexibility, especially since most calculators are equipped to find common and natural logarithms.
- For example, to evaluate \( \log_{9}(27) \), you might use base 10. So, \( \log_{9}(27) = \frac{\log_{10}(27)}{\log_{10}(9)} \).
- Calculate each part using a calculator, \( \log_{10}(27) \approx 1.431 \) and \( \log_{10}(9) \approx 0.954 \).
- Dividing these results gives approximately 1.5, confirming the value of \( \log_{9}(27) \).
By using this formula, you can simplify logarithmic expressions to a form easily computed with basic calculators.

Exponential Functions

Logarithms and exponential functions are closely linked. Understanding one helps to grasp the other as they are inverse operations. For example, if you know \( a^c = b \), then \( \log_{a}(b) = c \).
This relationship is crucial because:

• Exponential functions grow by constant factors, while logarithms grow at a decreasing rate.
• The exponential form is practical in solving logarithms analytically, as demonstrated when solving \( \log_9(27) \). We rewrote bases to \( 9^x = 27 \) leading to \( (3^2)^x = 3^3 \) becoming \( 3^{2x} = 3^3 \) and then solving \( 2x = 3 \).

Getting comfortable with switching between these forms enhances your problem-solving skills with exponential growth and decay scenarios.

Graph of Logarithms

Visualizing how logarithms behave through their graphs can solidify your understanding of their nature. A logarithmic graph, like that of \( y = \log_{9}(x) \), shows a curve that increases slowly and continuously.
Key characteristics include:

• The graph passes through the point (1,0), as \( \log_{a}(1) = 0 \) for any base \( a \).
• As x increases, y increases, which reflects the logarithmic scaling.
• The increasing rate flattens as x becomes large, appearing as though it approaches a horizontal asymptote.

To corroborate results, check points on this graph such as the one found for \( x=27 \). The point where the graph intersects x=27 should align with your calculated \( y \) value, confirming the logarithmic result. This visual guide backs up analytical solutions, offering an intuitive check of calculations.