Question

Consider two logarithmic functions: f(x)=log3 x
and g(x)=log1/3 x


Determine the coordinates (x,y)
of the point where f(x)
and g(x)
intersect.

Answer 1

To determine the coordinates of the point where the two logarithmic functions \( f(x) = \log_3 x \) and \( g(x) = \log_{\frac{1}{3}} x \) intersect, we need to set them equal to each other:

\[
\log_3 x = \log_{\frac{1}{3}} x
\]

We can use the change of base formula to express \( g(x) \) in terms of base 3:

\[
g(x) = \log_{\frac{1}{3}} x = \frac{\log_3 x}{\log_3 \frac{1}{3}}
\]

Since \( \frac{1}{3} = 3^{-1} \), we have:

\[
\log_3 \frac{1}{3} = -1
\]

Thus, we can rewrite \( g(x) \):

\[
g(x) = \frac{\log_3 x}{-1} = -\log_3 x
\]

Now, substituting this back into the intersection equation:

\[
\log_3 x = -\log_3 x
\]

Adding \( \log_3 x \) to both sides gives:

\[
2\log_3 x = 0
\]

Dividing both sides by 2:

\[
\log_3 x = 0
\]

This means:

\[
x = 3^0 = 1
\]

Now, let's find the corresponding \( y \) value by substituting \( x = 1 \) into either function. We can use \( f(x) \):

\[
f(1) = \log_3 1 = 0
\]

Thus, the coordinates of the intersection point are:

\[
(x, y) = (1, 0)
\]

In conclusion, the point where \( f(x) \) and \( g(x) \) intersect is:

\[
\boxed{(1, 0)}
\]