Question

If the exponential function f(x)=9
x
includes the point (2,81) on the graph of the exponential curve, which point can be described as lying on the graph of the logarithmic function g(x)=log
9
​
x?
A. (1,9)
B. (81,2)
C. (2,81)
D. (1,0)

Answer 1

To find out which point lies on the graph of the logarithmic function \( g(x) = \log_9 x \), we need to use the relationship between exponential and logarithmic functions.

Given that the function \( f(x) = 9^x \) includes the point \( (2, 81) \), we can state:

\[
f(2) = 9^2 = 81
\]

This means that \( f(2) = 81 \) is correct.

Now, for the logarithmic function \( g(x) = \log_9 x \), we know that:

\[
g(a) = b \text{ if and only if } 9^b = a.
\]

Since we have \( 81 = 9^2 \), we can express this in terms of the logarithmic function:

\[
g(81) = \log_9 81 = 2.
\]

This tells us that the point \( (81, 2) \) lies on the graph of the logarithmic function.

Now, let's look at the answer choices:

A. \( (1,9) \)
B. \( (81,2) \)
C. \( (2,81) \)
D. \( (1,0) \)

Since we determined that \( (81, 2) \) is the point that lies on \( g(x) \), the correct answer is:

**B. (81,2)**