Question

If the exponential function f(x)=9x
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)=log9x
?

Answer 1

To determine a point on the graph of the logarithmic function \( g(x) = \log_9{x} \), we can use the point given on the exponential function \( f(x) = 9^x \).

Since the point \( (2, 81) \) lies on the graph of \( f(x) \), we know that:

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

The relationship between exponential functions and logarithmic functions is that if \( f(a) = b \), then \( g(b) = a \). Here, we have:

- \( a = 2 \)
- \( b = 81 \)

So, we can write:

\[
g(81) = \log_9{81}
\]

Now, let's evaluate \( \log_9{81} \). Since \( 81 = 9^2 \), we can express the logarithm as:

\[
\log_9{81} = \log_9{(9^2)} = 2
\]

This means that the point \( (81, 2) \) lies on the graph of the logarithmic function \( g(x) \).

Thus, the point that can be described as lying on the graph of the logarithmic function \( g(x) = \log_9{x} \) is:

\[
\boxed{(81, 2)}
\]