Problem 55 - A Graphical Approach to College Algebra

Question

Evaluate each logarithm in three ways: (a) Use the definition of logarithm in Section 5.3 to find the exact value analytically. (b) Support the result of part (a) by using the change-of-base rule and common logarithms on your calculator. (c) Support the result of part (a) by locating the appropriate point on the graph of the function y=\log _{a} x. $$\log _{16}\left(\frac{1}{8}\right)$$

Step-by-Step Solution

Answer

The value is \(-\frac{3}{4}\).

Step 1: Use the definition of logarithm

The definition of a logarithm states that if \( \log_a b = c \), then \( a^c = b \). For the given logarithm \( \log_{16}\left(\frac{1}{8}\right) \), we need to find \( c \) such that \( 16^c = \frac{1}{8} \). We can express both 16 and \( \frac{1}{8} \) as powers of 2: \( 16 = 2^4 \) and \( \frac{1}{8} = 2^{-3} \). Therefore, \( 2^{4c} = 2^{-3} \). By equating the exponents, we get \( 4c = -3 \). Solving for \( c \), we find \( c = -\frac{3}{4} \). Thus, \( \log_{16}\left(\frac{1}{8}\right) = -\frac{3}{4} \).

Step 2: Use the change-of-base formula

The change-of-base formula is \( \log_a b = \frac{\log_c b}{\log_c a} \). We can use common logarithms (base 10) for this step. Applying the formula: \( \log_{16}\left(\frac{1}{8}\right) = \frac{\log_{10}\left(\frac{1}{8}\right)}{\log_{10}16} \). Calculate \( \log_{10}\left(\frac{1}{8}\right) \) using a calculator to get approximately \( -0.903 \), and \( \log_{10}16 \) to get approximately \( 1.204 \). Thus, the expression becomes \( \frac{-0.903}{1.204} \approx -0.749 \), which supports the result \( -\frac{3}{4} \).

Step 3: Use the graph of \(y = \log_a x\)

The graph of \( y = \log_{16} x \) assists us in visualizing this logarithm. You need to identify the graph of the function and locate where \( x = \frac{1}{8} \). Notice that the y-value obtained from this x-value should correspond to \( y = -\frac{3}{4} \), just like our previous analytical result. This step makes sure visually that when \( x \) changes as \( \frac{1}{8} \), the logarithmic function returns a value of \( -\frac{3}{4} \), confirming our analytical solution.

Key Concepts

Change-of-base formula

The change-of-base formula is a helpful tool for evaluating logarithms when the base is not suitable for direct calculation with a standard calculator. Generally, calculators provide logarithm functions only for bases 10 (common logarithm) and e (natural logarithm). To convert a logarithm with another base to these standard bases, we use the formula:\[ \log_a b = \frac{\log_c b}{\log_c a} \]This formula essentially states that you can take the logarithm of the number and the base in terms of a new, more manageable base (like 10 or e), and then divide the two results.

• For example: To evaluate \( \log_{16}\left(\frac{1}{8}\right) \), we apply the change-of-base formula using base 10: \( \log_{16}\left(\frac{1}{8}\right) = \frac{\log_{10}\left(\frac{1}{8}\right)}{\log_{10}16} \).
• By computing these values using a calculator, \( \log_{10}\left(\frac{1}{8}\right) \approx -0.903 \) and \( \log_{10}16 \approx 1.204 \), this results in \( \frac{-0.903}{1.204} \approx -0.749 \).

This calculated value supports our analytical solution and confirms that the change-of-base formula is valid and useful, especially when dealing with logarithms with unconventional bases.

Graphical interpretation of functions

Visualizing a logarithmic function helps grasp its behavior and relationships with the underlying numbers. The graph of a logarithmic function \( y = \log_a x \) is essential in understanding how the function's output is affected by changes in \( x \). For bases greater than 1, the graph of the logarithm function will:

• Curve upwards as \( x \) increases, passing through the point \((1, 0)\).
• Creep towards negative infinity when \( x \) approaches zero from the positive side.

To find \( \log_{16}(\frac{1}{8}) \) by the graph, look for the point where \( x = \frac{1}{8} \). According to the function, the \( y \)-value here should match our calculated logarithm value, which is \(-\frac{3}{4}\).
By locating this point on the graph, the graph visually confirms the logarithm value we found analytically. It shows us that even as the base and the number change, the relationship maintained by a logarithmic function holds true. Visual patterns in graphs often provide deeper insights that support analytical solutions.

Logarithmic function properties

Logarithms have several vital properties that make working with them easier and more intuitive. These properties help solve various logarithmic equations and expressions. Some of the key logarithmic function properties include:

• Product Rule: \( \log_a(mn) = \log_a (m) + \log_a(n) \)
• Quotient Rule: \( \log_a\left(\frac{m}{n}\right) = \log_a(m) - \log_a(n) \)
• Power Rule: \( \log_a(m^n) = n \cdot \log_a(m) \)
• Change of Base Formula: Use the formula \( \log_a b = \frac{\log_c b}{\log_c a} \) to switch bases

These rules are useful for simplifying logarithmic expressions, finding equivalent values, and solving equations. For instance, knowing that \( \log_{16}\left(\frac{1}{8}\right) \) can be expressed in terms of the powers of 2 allowed us to equate the exponents and solve for the logarithm without using a calculator.
Understanding and applying these properties can make the manipulation and comprehension of logarithmic functions much easier, aligning with both analytical and graphical methods for solving logarithms.