Question

At what point do the graphs of y= log5 x and y= log0.5 x

intersect?

A
(1, 0)

B
(1, 2.5)

C
(0, 1)

D
(0.5, -5)

Answer 1

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

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

Using the change of base formula, we can rewrite both logarithms in terms of natural logarithms:

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

Setting these equal gives us:

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

Assuming \( \ln x \neq 0 \) (which means \( x \neq 1 \)), we can cancel \( \ln x \) from both sides:

\[
\frac{1}{\ln 5} = \frac{1}{\ln 0.5}
\]

This means:

\[
\ln 0.5 = \ln 5
\]

This is not true, therefore we need to consider the case when \( \ln x = 0 \):

\[
\ln x = 0 \quad \Rightarrow \quad x = 1.
\]

Now, substituting \( x = 1 \) back into either logarithmic function to find \( y \):

\[
y = \log_5(1) = 0
\]

And also,

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

Thus, the point of intersection is:

\[
(1, 0).
\]

Now we'll check the options given:

A. \( (1, 0) \)
B. \( (1, 2.5) \)
C. \( (0, 1) \)
D. \( (0.5, -5) \)

The correct answer is:

**A. (1, 0)**.