Exponential and Logarithmic Functions
Exponential and Logarithmic Functions tests exponent laws, logarithm laws, equations and inequalities, and common and natural logarithms. Domain restrictions are central because logarithms accept only positive arguments.
Core rules
- a^m a^n=a^{m+n}, (a^m)^n=a^{mn}, a^{-n}=1/a^n.
- log_a(xy)=log_a x+log_a y; log_a(x/y)=log_a x-log_a y; log_a x^k=k log_a x.
- log_a b=log_c b/log_c a. Common log is log_10, natural log is ln=log_e.
- For a>1, log_a x is increasing; for 0<a<1, it is decreasing, so inequalities reverse when the base is less than 1.
- Converting between exponential and logarithmic form, y=a^x iff x=log_a y, is often the fastest first step.
Worked example
Solve log_2(x-1)+log_2(x-3)=3. Domain: x>3. Combine: log_2((x-1)(x-3))=3, so (x-1)(x-3)=8. Then x^2-4x-5=0, giving x=5 or x=-1. Only x=5 satisfies x>3, so the solution is x=5.
Traps
Always state domain restrictions before combining logs. Do not write log(x+y)=logx+logy. When solving exponential inequalities, reversing the inequality depends on the base, not on the exponent. Check extraneous roots created by combining or transforming logarithmic equations.