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An exponential equation is an equation with a variable exponent or an exponent with a variable in it. For example: $$2^x=256$$ and $$3^{2x-4}=342$$ are both exponential equations.
We can solve exponential equations in one of two ways, depending on the bases of the equation's terms.

Solving exponential equations using logarithms
The first way to solve exponential equations does not take the bases into account and involves using the following logarithmic rule to move and isolate the equation's variable:

$${\log }_x{a}^b=b\cdot {\log }_{x}a$$

Finding the log of a number with a variable as an exponent allows us to move the exponent to the front of the equation, making it a multiplier on the log. From there, we can isolate the variable and solve the equation.

See an example problem here

Solving exponential equations using exponent properties
The second way to solve exponential equations uses properties of exponents. If we can get both sides of the equation to have the same base, then we can set the exponents equal to each other. This relationship can be expressed as:
if $$x^a=x^b$$ then $$a=b$$

for example:
$$2^x=256$$
Because $$256=2^8$$ then $$2^x=2^8$$, meaning $$x=8$$.