Question
At what point do the graphs of y=log5 x (5 is the base of x) and y= log0.5 x(0.5 is the base of x) intersect? What is the y-coordinate?
Answers
GPT 3.5
The graphs of y = log5 x and y = log0.5 x intersect when the two equations are equal to each other:
log5 x = log0.5 x
Using the property of logarithms that states log_a x = log_b x / log_b a, we can rewrite the equation as:
log x / log 5 = log x / log 0.5
Simplifying further:
log x / log 5 = log 2 / log 2
Since log 2 / log 2 equals 1, we can rewrite the equation as:
log x / log 5 = 1
Solving for x:
x = 5
Therefore, the graphs of y = log5 x and y = log0.5 x intersect at x = 5. To find the y-coordinate, we can substitute x = 5 into either of the equations. Let's use y = log5 x:
y = log5 5
y = 1
Therefore, the graphs intersect at the point (5, 1), where the y-coordinate is 1.
log5 x = log0.5 x
Using the property of logarithms that states log_a x = log_b x / log_b a, we can rewrite the equation as:
log x / log 5 = log x / log 0.5
Simplifying further:
log x / log 5 = log 2 / log 2
Since log 2 / log 2 equals 1, we can rewrite the equation as:
log x / log 5 = 1
Solving for x:
x = 5
Therefore, the graphs of y = log5 x and y = log0.5 x intersect at x = 5. To find the y-coordinate, we can substitute x = 5 into either of the equations. Let's use y = log5 x:
y = log5 5
y = 1
Therefore, the graphs intersect at the point (5, 1), where the y-coordinate is 1.