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Some of the content of this guide was modeled after a guide originally created by the Openstax and has been adapted for the GPRC Learning Commons in October 2020. The graphs are generated using Desmos. This work is licensed under a Creative Commons BY 4.0 International License

Solving Problems Involving Exponential Equations


In some cases, we have to solve equations that include an exponential function where the base of the function is the variable. 


Example: Solve 
First, we have to cancel the coefficient behind the exponential function. Therefore, we divide both sides by 5:

We need to isolate x, to do so, we have to cancel the power of x:


In some other cases, one has to write an exponential function in a different base. 


Example: Convert  to base 
We substitute into the original equation:

we multiply the exponents on the right-hand-side of the equation:


Example: Solve 
The base of the exponential function on the left-hand-side is 3, therefore, we try to write the right-hand-side in base 3:

since the bases on the right-hand-side are the same, we add their exponents:

the bases are equal on both sides of the equation, so, the exponents are also equal:








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