at what point do the graphs of $y = \\log_{5} x$ and $y = \\log_{0.5} x$ intersect? (5 points)○ (0.5, -5)○…
at what point do the graphs of $y = \\log_{5} x$ and $y = \\log_{0.5} x$ intersect? (5 points)○ (0.5, -5)○ (0, 1)○ (1, 2.5)○ (1, 0)
Answer
Explanation:
Step1: Set equations equal
$\log_{5} x = \log_{0.5} x$
Step2: Use change of base formula
$\frac{\ln x}{\ln 5} = \frac{\ln x}{\ln 0.5}$
Step3: Rearrange to solve for $x$
$\ln x \left(\frac{1}{\ln 5} - \frac{1}{\ln 0.5}\right) = 0$ Since $\frac{1}{\ln 5} - \frac{1}{\ln 0.5} \neq 0$, $\ln x = 0$, so $x=1$.
Step4: Find $y$-value
Substitute $x=1$ into $y=\log_{5} x$: $y=\log_{5} 1 = 0$
Answer:
D. (1, 0)