Question

At what point do the graphs of y=log5 x
and y=log0.5 x
intersect?(1 point)
Responses

(0, 1)
(0, 1)

(1, 2.5)
(1, 2.5)

(0.5, -5)
(0.5, -5)

(1, 0)
(1, 0)

Answer 1

To find the point of intersection of the graphs of \( y = \log_5 x \) and \( y = \log_{0.5} x \), we need to set the two equations equal to each other:

\[
\log_5 x = \log_{0.5} x
\]

Using the change of base formula for logarithms, we can rewrite both sides:

\[
\frac{\log x}{\log 5} = \frac{\log x}{\log 0.5}
\]

Next, we cross-multiply to eliminate the fractions:

\[
\log x \cdot \log 0.5 = \log x \cdot \log 5
\]

Assuming that \( \log x \neq 0 \) (i.e., \( x \neq 1 \)), we can simplify by dividing both sides by \( \log x \):

\[
\log 0.5 = \log 5
\]

This statement is not true, so we consider the case \( \log x = 0 \), which occurs at \( x = 1 \):

\[
\log_5(1) = 0 \quad \text{and} \quad \log_{0.5}(1) = 0
\]

Both equal to 0, implying \( (1, 0) \) is indeed a point of intersection.

Thus, the graphs of \( y = \log_5 x \) and \( y = \log_{0.5} x \) intersect at the point:

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