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Exponential and Logarithmic Functions

Solving Exponential Equations Graphically and Algebraically

Exponential equations — equations in which the independent variable is an exponent — can be solved graphically and algebraically. Depending on the equation, different algebraic approaches can be used.

Method

Solving Exponential Equations Graphically

If the dependent variable of an exponential function written in the form
is exchanged for a constant, say C, the result is a one-variable equation:
This type of equation is called an exponential equation, and can be solved graphically. This is done by first graphing the function then finding the x-coordinate of the point(s) on the graph with the y-coordinate C. The x-coordinate(s) is the solution to the equation.

Example

Use the graph to solve the exponential equation

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Use the graph to solve the equation

Show Solution expand_more
The graph shows all x-y points that satisfy the function rule Let's compare the function rule and the equation.
The only difference between these two equalities is that the independent variable, y, is replaced by a 3 in the equation. Thus, we solve the equation by finding the x-coordinate of any point on the graph that has the y-coordinate 3.

We can identify one such point in the graph. Let's now find the x-coordinate of this point graphically.

This x-coordinate is not easily read from the graph, so we'll have to make an approximation. It's just a bit bigger than 3, so we'll use 3.1. This means that an approximate solution to the equation is We can verify this by substituting it into equation to see if a true statement is made.

The right-hand side and the left-hand side are approximately equal, so we have indeed found an approximate solution to the equation:

Method

Solving Exponential Equations Algebraically

Let b be a positive number other than 1 and a(x) and c(x) be two algebraic expressions in terms of the same variable. If an exponential equation is or can be written in the following form, then it can be solved algebraically by using the Property of Equality for Exponential Equations.

Consider an example exponential equation.
To solve the equation, four steps must be followed.
1
Rewrite the Expressions on Both Sides as Powers With the Same Base
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First, the exponential expressions on both sides must be rewritten as powers with the same base. In this case, 4096 can be written as a power of 4.
If the exponential expressions on both sides of the equation already have the same base, this step can be skipped.
2
Use the Property of Equality for Exponential Equations
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Next, the Property of Equality for Exponential Equations can be used. Since the bases are equal, the exponents must be equal for the equation to be true.
3
Solve the Resulting Equation
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The resulting equation can be solved for the variable.
4
Verify the Solution
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Finally, the obtained solution can be verified by substituting it into the given equation.
Evaluate left-hand side
Since a true statement was obtained, x=3 is a solution to the equation. It is important to verify all the obtained solutions, since sometimes this method can lead to extraneous solutions.

Example

Solve the exponential equation

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Solve the equation
Show Solution expand_more
To begin, notice that both sides of the equation are exponential expressions with base 3. Since they have the same base, the exponents must be equal. This gives the equivalent equation
5x3=-2x+11,
which we can solve using inverse operations.
5x3=-2x+11
7x3=11
7x=14
x=2
The solution to the equation is x=2.
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