Question

At what point do the graphs of y=log5xand y=log0.5xintersect?(1 point)Responses(1,0)left parenthesis 1 comma 0 right parenthesis(1,2.5)left parenthesis 1 comma 2.5 right parenthesis(0.5,−5)left parenthesis 0.5 comma negative 5 right parenthesis(0,1)left parenthesis 0 comma 1 right parenthesis

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:

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

We can use the change of base formula to rewrite both logarithms in terms of base \( 10 \) (or any other base):

\[
\log_5 x = \frac{\log_{10} x}{\log_{10} 5}
\]
\[
\log_{0.5} x = \frac{\log_{10} x}{\log_{10} 0.5}
\]

Thus, we can rewrite the equation:

\[
\frac{\log_{10} x}{\log_{10} 5} = \frac{\log_{10} x}{\log_{10} 0.5}
\]

We can multiply both sides of the equation by \( \log_{10} 5 \cdot \log_{10} 0.5 \) (assuming \( \log_{10} x \neq 0 \)) to get:

\[
\log_{10} x \cdot \log_{10} 0.5 = \log_{10} x \cdot \log_{10} 5
\]

Now, if \( \log_{10} x \neq 0 \), we can divide both sides by \( \log_{10} x \):

\[
\log_{10} 0.5 = \log_{10} 5
\]

However, since \( \log_{10} 0.5 \) is actually negative, we need to check the case when \( \log_{10} x = 0 \). This occurs when:

\[
x = 1
\]

Now, let's evaluate both logarithms to see if they equal at this point:

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

Thus, both equations give \( y = 0 \) when \( x = 1 \).

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

\[
(1, 0)
\]

So, the answer is \( \boxed{(1, 0)} \).