Solving Exponential Equations Basic

Basic Techniques


Basic techniques for solving equations with exponentials

The two most useful techniques for solving equations involving exponentials are:
1.

Look for a common base; and

2.

Look for a common exponent.

The first enables you to manipulate terms in the equation using the rules , and . The second lets you take advantage of the rule . It is often the case that the two rules are used in tandem; for example, if then .



Example: Find such that .
Solution: Note that 8 and 4 can both be expressed as powers of 2 ( and , so , so .


Using Logarithms


See the lessons on logarithms elsewhere in this collection for information regarding the properties of logarithms.
Using logarithms to solve equations with exponentials

An equation which can be recast into the form can be solved for in terms of logarithms: .



Example: Find such that .
Solution: Since we have . Multiplying through by , we get
Taking the base3 logarithm of both sides, we have , so, finally, .


Explore what occurs when parameters are modified


The graph shows two exponential functions of the form . Use the sliders to change the parameters , , and 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 parameters changes from less than 1 to greater than 1, or when one of the parameters changes from positive to negative, and vice versa.


Download Help Document
Was this information helpful?