# Exponential Equations

Equations of the form a^(f(x))=a^(g(x)), where a>0,a!=1 are called exponential equations.

To solve such equation we use following property: exponential equation a^(f(x))=a^(g(x)) is equivalent to the equation f(x)=g(x).

There are two methods of solving of exponential equation:

1. Method of equating exponents, i.e. transformation of given equation into the form a^(f(x))=a^(g(x)) and then to the equivalent form f(x)=g(x).
2. Method of introducing new variable.

Example 1. Solve equation ((0.2)^(x-0.5))/(sqrt(5))=5*(0.04)^(x-1).

Let's transform all powers to the base 0.2: we have that sqrt(5)=sqrt((1/5)^(-1))=sqrt((0.2)^(-1))=(0.2)^(-0.5), 5=(0.2)^(-1), (0.04)^(x-1)=((0.2)^2)^(x-1)=(0.2)^(2(x-1)).

Therefore, equation can be rewritten as ((0.2)^(x-0.5))/((0.2)^(-0.5))=(0.2)^(-1)*(0.2)^(2(x-1)) or (0.2)^(x-0.5-(-0.5))=(0.2)^(-1+2(x-1)).

After simplifying we obtain that (0.2)^x=(0.2)^(2x-3). This equation is equivalent to the equation x=2x-3, from which x=3.

Example 2. Solve equation 4^x+2^(x+1)-24=0.

Since 4^x=(2^2)^x=(2^x)^2 and 2^(x+1)=2*2^x then equation can be rewritten as (2^x)^2+2*2^x-24=0.

Now we introduce new variable: let y=2^x then y^2+2y-24=0. This equation has two roots: y=4 and y=-6.

Thus, we obtained set of equations: 2^x=4,\ 2^x=-6.

From first equation x=2, second equation doesn't have roots, because 2^x>0 for any x.

Therefore, initial equation has only one root: x=2.