Solve an Exponential Equation: Relating the Bases

Problem: Solve the exponential equation.

$36^{x}=(1/6)^{x-3}$

Solution:

The exponential equation	$36^{x}=(1/6)^{x-3}$
The bases of the exponents on each side of the equation can be made the same. 36 can be written as and can be written as $6^{-1}$	$36^{x}=(1/6)^{x-3}$ $(6^2)^{x}=(6^{-1})^{x-3}$
Use the power rule for exponents to multiply the exponents. Power Rule for Exponents $(b^m)^n=b^{mn}$	$(6^2)^{x}=(6^{-1})^{x-3}$ $6^{2x}=6^{-1(x-3)}$ $6^{2x}=6^{-x+3}$
Exponential functions are one-to-one thus giving us the property that if the bases are the same the exponents are equal. If then	$6^{2x}=6^{-x+3}$
Solve the remaining equation. This equation is linear, first get the variables to the same side.
Solve: Get the variable by itself.
Check: Substitute into the original equation.	$36^{1}=(1/6)^{1-3}$ $36=(1/6)^{-2}$ $36=6^{2}$