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?
The graphs of y = log5 x and y = log0.5 x intersect at one point when x = 1.
To see this, we can solve for x when the two equations are equal:
log5 x = log0.5 x
Since the bases are different, we can use the property of logarithms that states loga x = logb x / logb a.
So we have:
log5 x = log x / log5 = log0.5 x
Now we can equate the two expressions:
log x / log5 = log x / log(1/2)
Multiplying both sides by log5, we get:
log x = log x log(1/2)/log5
Since log a b = log a + log b, we can rewrite the right side as:
log x = log x (log1-log2) / log5
log x = logx / log5
This tells us that log x = log x, or x = 1.
Therefore, the intersection point of the two graphs occurs at x = 1.