Questions: The function f(x)=log9(x) is the logarithm function with base 9. So f(9)= 1, f(1)= 0, f(1/9)= -1, f(81)= 2, and f(3)= 1/2.

The function f(x)=log9(x) is the logarithm function with base 9. So f(9)= 1, f(1)= 0, f(1/9)= -1, f(81)= 2, and f(3)= 1/2.
Transcript text: The function $f(x)=\log _{9}(x)$ is the logarithm function with base $\square$ .So $f(9)=$ $\square$ , $f(1)=$ $\square$ , $f\left(\frac{1}{9}\right)=$ $\square$ , $f(81)=$ $\square$ , and $f(3)=$ $\square$ .
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Solution

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Solution Steps

To solve the given problem, we need to evaluate the logarithm function \( f(x) = \log_{9}(x) \) for different values of \( x \). The base of the logarithm is 9. We will use the properties of logarithms to find the values of \( f(9) \), \( f(1) \), \( f\left(\frac{1}{9}\right) \), \( f(81) \), and \( f(3) \).

Solution Approach
  1. Identify the base of the logarithm, which is 9.
  2. Use the definition of logarithms to evaluate \( f(x) \) for the given values of \( x \).
Step 1: Evaluate \( f(9) \)

Using the definition of logarithms, we have: \[ f(9) = \log_{9}(9) = 1 \]

Step 2: Evaluate \( f(1) \)

For the logarithm of 1, we find: \[ f(1) = \log_{9}(1) = 0 \]

Step 3: Evaluate \( f\left(\frac{1}{9}\right) \)

Using the property of logarithms, we calculate: \[ f\left(\frac{1}{9}\right) = \log_{9}\left(\frac{1}{9}\right) = -1 \]

Step 4: Evaluate \( f(81) \)

Since \( 81 = 9^2 \), we have: \[ f(81) = \log_{9}(81) = \log_{9}(9^2) = 2 \]

Step 5: Evaluate \( f(3) \)

Using the change of base formula: \[ f(3) = \log_{9}(3) = \frac{\log_{10}(3)}{\log_{10}(9)} \approx 0.5 \]

Final Answer

The values are:

  • \( f(9) = 1 \)
  • \( f(1) = 0 \)
  • \( f\left(\frac{1}{9}\right) = -1 \)
  • \( f(81) = 2 \)
  • \( f(3) \approx 0.5 \)

Thus, the final answers are: \[ \boxed{f(9) = 1, \quad f(1) = 0, \quad f\left(\frac{1}{9}\right) = -1} \]

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