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Tutorials that teach
Solving an Exponential Equation

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- Exponential Equations with the Same Base
- Rewriting the Base

When solving for any type of equation, a sound strategy is to apply inverse operations to undo operations being performed to the variable. Doing so isolates the variable to one side of the equation, where we can then evaluate the other side to find the our solution.

In general, exponential equations can be solved by applying a logarithm to both sides of the equation. This is because logarithms and exponents are inverse operations. However, we explore this method in a different lesson. In this lesson, we focus on how to analyze the bases involved in the equations. If the bases are the same, or if they can be rewritten to match, we can actually solve exponential equations without using logarithms.

**Exponential Equations with the Same Base**

Consider the following equation:

In both exponential expressions, the base is 6. This means that the same quantity is the exponent to 6 on both sides of the equation, therefore 2x + 9 and –5x + 2 must be equal quantities. We can create an equivalent equation that is actually linear in nature, and solve for x:

If the bases are the same in exponential equations, we can set the exponents equal to each other, and isolate the variable as we normally do with other equations.

**Rewriting the Base**

When we are working with exponential equations in which the base numbers are not the same, it may appear as though we cannot solve using the strategy described in the section above. However, by closely examining the base numbers, we may be able to rewrite one or more of the bases in order to create an equivalent equation with common bases. If we can do this, we can solve the equation using a similar strategy as before.

When we use this strategy, we will need to apply the power of a power property of exponents. This property allows us to multiply exponents in cases where a base number is raised to an exponent power, and then raised to an exponent power again.

Consider the following equation:

At first glance, it may appear as though we cannot solve this equation using our strategy from before. However, we notice that both 4 and 8 are powers of 2. That is, 4 is the same as 2^{2} and 8 is the same as 2^{3}. Let's make these substitutions in our equation.

We can multiply the two exponents on each side of the equation using the power of powers property of exponents, and then solve as we did in the first example.