Things I know about logarithms and graphing rational equations

I'm doing this without my book. Honest!

First, logarithms:

• A logarithm is an exponent.


• If log _ax=y, then a^y=x


• log _aa = 1


• log _a1 = 0


• a^log_ax = x


• if log _ax = log _ay, then x=y


• log _ax = log ₁₀x / log ₁₀a

Next, graphing rational equations

• To find the y intercepts, solve the equation for x=0


• To find the x intercepts, find the zeroes of the numerator


• The vertical asymptote is the zero of the denominator


• The horizontal asymptote is determined under the following conditions:


1. If the highest degree of the numerator is > the highest degree of the denominator, the H.A. is 1


2. If the highest degree of the numerator is = the highest degree of the denominator, the H.A. is the leading coefficient of the numerator/leading coefficient of the denominator


3. If the highest degree of the numerator is < the highest degree of the denominator, there is no H.A.


• If the degree of the numerator is exactly one more than the degree of the denominator, the slant asymptote is the line ay=bx, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.

I'll update this to reflect any mistakes I've made...

Update: I was mostly wrong about finding the horizontal asymptote:

1. If the highest degree of the numerator is > the highest degree of the denominator, the graph has no H.A.


2. If the highest degree of the numerator is = the highest degree of the denominator, the H.A. is the leading coefficient of the numerator/leading coefficient of the denominator (I got this one right).


3. If the highest degree of the numerator is < the highest degree of the denominator, the H.A. is the x-axis