# Exponential Inequalities

When we solve inequalities of the form a^(f(x))>a^(g(x)) we need to remember, that exponential function y=a^x is increasing when a>1 and decreasing when 0<a<1.

Therefore, when a>1, a^(f(x))>a^(g(x)) is equivalent to the inequality f(x)>g(x). If 0<a<1, then a^(f(x))>a^(g(x)) is euqivalent to f(x)<g(x).

Example 1. Solve 2^(3x+7)<2^(2x-1).

Here, base of exponent is greater than 1 (it is 2), therefore, equivalent inequality is 3x+7<2x-1. Solving it, we obtain that x<-8.

Example 2. Solve (0.04)^(5x-x^2-8)<=625.

Since 625=25^2=((1/(25))^-1)^2=(1/(25))^(-2)=(0.04)^(-2), then we can rewrite inequality as (0.04)^(5x-x^2-8)<=(0.04)^-2.

Since 0<0.04<1, then equivalent inequality is 5x-x^2-8>=-2. From this we obtain that

-x^2+5x-6>=0, x^2-5x+6<=0, (x-2)(x-3)<=0.

Using method of intervals we obtain that solution is 2<=x<=3.