91Ó°ÊÓ

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In our exercise, the exponential function given is \( y = \left(\frac{3}{2}\right)^x \). Here, \( \frac{3}{2} \) is the base and \( x \) is the exponent. Exponential functions have a few key properties:

• Growth and Decay: If the base is greater than 1, the function represents exponential growth. If the base is between 0 and 1, it represents exponential decay.
• Y-intercept: At \( x = 0 \), the value of any exponential function is always 1 (since anything raised to the power of 0 is 1).
• Asymptote: The x-axis (or y = 0) is a horizontal asymptote, meaning the graph gets infinitely close to the x-axis but never actually touches it as \( x \) goes to negative infinity.

When graphing this function, start by creating a table of values for specific \( x \) values, like -2, -1, 0, 1, and 2. Calculate the corresponding \( y \) values to help plot the points accurately.
For example:

• If \( x = -1 \), then \( y = \left(\frac{3}{2}\right)^{-1} = \frac{2}{3} \).
• If \( x = 1 \), then \( y = \left(\frac{3}{2}\right)^1 = 1.5 \).

A logarithmic function is the inverse of an exponential function. For our problem, the logarithmic function provided is \( y = \log_{3/2} x \). This function means that \( y \) is the exponent to which the base \( \frac{3}{2} \) must be raised to produce \( x \). Key properties of logarithmic functions include:

• Domain and Range: The domain of a logarithmic function is all positive real numbers, and the range is all real numbers.
• X-intercept: At \( x = 1 \), \( y \) is always 0 since the logarithm of 1 (regardless of the base) is 0.
• Asymptote: The y-axis (or x = 0) serves as a vertical asymptote, as the function approaches negative infinity.

To plot this function effectively, you need to select a range of positive \( x \) values, such as 0.5, 1, 1.5, 2, and 3. Then, using the change of base formula, compute the corresponding \( y \) values:
\[ y = \frac{\log(x)}{\log(\frac{3}{2})} \].
For example:

• If \( x = 2 \), then \( y = \log_{3/2}(2) = \frac{\log(2)}{\log(\frac{3}{2})} \approx 1.7095 \).
• If \( x = 0.5 \), then \( y = \log_{3/2}(0.5) = \frac{\log(0.5)}{\log(\frac{3}{2})} \approx -1 \).

The change of base formula is an essential tool for evaluating logarithms where the base is not 10 (common logarithm) or \( e \) (natural logarithm). It states that: \[ \log_b a = \frac{\log a}{\log b} \]
This formula helps us convert a logarithm with any base into a common logarithm or natural logarithm.
Let's apply this to our logarithmic function from the exercise, \( y = \log_{3/2} x \): \[ \log_{3/2} x = \frac{\log x}{\log (\frac{3}{2})} \]
Here, you can easily use a calculator to find both \( \log x \) and \( \log (\frac{3}{2}) \), allowing for the accurate graphing of the logarithmic function.

• Step-by-Step Application: Pick a few values of \( x \), then use the change of base formula to find \( y \). For instance, if \( x = 3 \), then \( y = \log_{3/2} 3 = \frac{\log 3}{\log (\frac{3}{2})} \approx 2.7095 \).
• Common and Natural Logarithms: The change of base formula is versatile and works for \( \log_{10} \) and \( \log_{e} \), making it helpful for various base conversions.

By understanding this concept, you can more easily evaluate and graph complex logarithmic functions.