Solving Exponential Equations - Basic - Maple Help
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Solving Exponential Equations- Basic

Main Concept

Basic techniques for solving equations with exponentials

The two most useful techniques for solving equations involving exponentials are:



Look for a common base; and


Look for a common exponent.

The first enables you to manipulate terms in the equation using the rules axay=ax+y,  axay=axy and axy=axy. The second lets you take advantage of the rule axbx=abx. It is often the case that the two rules are used in tandem; for example, if c=ab then axcx=axabx=axaxbx=bx.



Example: Find x such that 82 x45 x=16.

Solution: Note that 8 and 4 can both be expressed as powers of 2 (8=23 and 4=22), so 82 x45 x=232 x225 x=26 x210 x=24 x=24x=16x=16, so x=1.

Using logarithms to solve equations with exponentials

See the lessons on logarithms elsewhere in this collection for information regarding the properties of logarithms.


An equation which can be recast into the form ax=b can be solved for x in terms of logarithms: x=logab.



Example: Find x such that 33 x=179x.

Solution: Since 9=32 we have 33 x=179x=1732x=1732 x. Multiplying through by 32 x, we get


33 x32 x=33 x+2 x=35 x=17 


Taking the base-3 logarithm of both sides, we have 5 x=log317, so, finally, x=15log317.

The graph shows two exponential functions of the form carx. Use the sliders to change the parameters c, a, and r for each function. The point of intersection of the graphs of the two functions, which is the point at which the two functions are equal, is shown as a magenta dot. Experiment with the sliders to see how the shapes of the graphs, and hence their points of intersection, are affected by each of the parameters. In particular, observe what happens when the value of one of the a parameters changes from less than 1 to greater than 1, or when one of the r parameters changes from positive to negative, and vice versa.





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