Expert: Sherry Wallin - 4/30/2010

Question
what is the x intercept when your dealing with:
y= Blog(baseC)AX  ?

Answer
Hi Tori~
    I prefer to write logarithms such as yours as: y = blog_c (ax) where the underscore '_' is used as a subscript for the base of the log. Also the argument is (ax) and should be in parentheses or else you are really saying [blog_c a]*x and I don't think that is what you mean. Moving on to how to find the x-intercept:

The x-intercept always occurs where y is 0. But if you make y = 0 then you have blog_c(ax) = 0 which says either b = 0 or log_c (ax) = 0 which is more difficult than the following technique:

Translate the logarithm into an exponential form: first I want to take the b outside the log and make it an exponent of (ax)^b so that we now have y = log_c(ax)^b and now use the rule to convert to exponential equation: c^y = (ax)^b and let y = 0 -> c^0 = (ax)^b -> 1 = (ax)^b. Now take the bth root of both sides: 1^(1/b) = [(ax)^b]^(1/b) -> 1 = ax and so it follows that x = 1/a which is the x intercept of the original equation. Now plug this back into the original equation and see the reasoning: 0 = blog_c(a(1/a)) -> 0 = blog_c 1 which is 0!

Math Prof