Question

At what point do the graphs of y=log5x and y=log0.5x intersect?(1 point) Responses (1,0) (1,0) (1,2.5) (1,2.5) (0,1) (0,1) (0.5, -5)

Answer 1

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

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

We can use the change of base formula for logarithms:

\[
\log_5 x = \frac{\log x}{\log 5}
\]
\[
\log_{0.5} x = \frac{\log x}{\log 0.5}
\]

Setting these equal to each other gives:

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

Assuming \( \log x \neq 0 \) (which is valid since \( x > 0 \)), we can cross-multiply:

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

Subtracting \( \log x \cdot \log 5 \) from both sides results in:

\[
\log x \cdot \log 0.5 - \log x \cdot \log 5 = 0
\]

Factoring out \( \log x \):

\[
\log x (\log 0.5 - \log 5) = 0
\]

Since \( \log x \neq 0 \), we have:

\[
\log 0.5 - \log 5 = 0
\]
\[
\log 0.5 = \log 5
\]

This implies:

\[
0.5 = 5 \quad \text{(which is not true)}
\]

Thus we consider \( x = 1 \):

Substituting \( x = 1 \) into the original equations, we find:

\[
y = \log_5 1 = 0
\]
\[
y = \log_{0.5} 1 = 0
\]

This confirms that \( (1, 0) \) is indeed a point of intersection.

**Final answer: (1, 0)**