SOLVING EXPONENTIAL EQUATIONS
SOLVING EXPONENTIAL EQUATIONS
Note:
If you would like an in-depth review of exponents, the rules of exponents, exponential functions and exponential equations, click on exponential function under Algebra.
Problem 7.4b:
Answer:
The exact answer are 
and 
The
approximate answers are 
and 
Solution:
In order to solve this equation, we have to isolate the exponential term. Since we cannot easily do this in the equation's present form, let's tinker with the equation until we have it in a form we can solve.
Factor the left side of the equation 
The only way that a product can equal zero is if at least one of the factors
is zero.
Now we have an equation where the exponential term is isolated. Take the
natural logarithm of both sides of the equation 
Now let's look at the second factor,
Now we have a second equation where the exponential term is isolated. Take
the natural logarithm of both sides of the equation 
The exact answers are and 
The approximate answers are and
Check these answers in the original equation.
Check the solution by substituting
1.2527629685 in the original equation for x. If the left side of the
equation equals the right side of the equation after the substitution, you
have found the correct answer.
Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 1.2527629685 for x, then x=1.2527629685 is a solution.
Check the solution 
by substituting
-1.09861228867 in the original equation for x. If the left side of the
equation equals the right side of the equation after the substitution, you
have found the correct answer.
Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -1.09861228867 for x, then x=-1.09861228867 is a solution.
You can also check your answer by graphing 
(formed by subtracting the right side of the
original equation from the left side). Look to see where the graph crosses
the x-axis; that will be the real solution. Note that the graph crosses the
x-axis at two places: -1.09861228867 and 1.2527629685. This means that
-1.09861228867 and 1.2527629685 are the real solutions.
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