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May 23rd

May 23rd

# SOLVING EXPONENTIAL EQUATIONS

PLEASE NOTE THAT YOU CANNOT ALWAYS USE A CALCULATOR ON THE ACCUPLACER -
COLLEGE-LEVEL MATHEMATICS TEST! YOU MUST BE ABLE TO DO SOME PROBLEMS
WITHOUT A CALCULATOR!

The fol lowing are some examples of the types of exponential equations that we are going to solve in
this section
.

Strategy for Solving Exp onential Equations

• If necessary, isolate the exponential expression on one side of the equation with a coefficient
of 1.
• By definition, if M = N, then . Ex pressed differently , either

place the word log (base 10 is as sumed ) in front of the entire right side and
the entire left side of the exponential equation

or

place the word ln (base e is assumed) in front of the entire right side and the
entire left side of the exponential equation.

• Your exponential equation is now a logarithmic equation . With the help of the Power Rule you
can now
"free" the variable from its exponential position.
• Solve for the variable.

Problem 1:

Solve . Round to 4 decimal places .

Method 1:

Since the exponential expression is already isolated we'll place the word ln
(log base e) in front of the right side and in front of the left side

Using the Power Rule , we get

Please note that by the basic Logarithm Properties

Therefore, we have to divide both sides by

and using the calculator, we find that .

Method 2:

Let's do this problem again. This time we'll use the common logarithm , that is
log base 10.

Now, place the word log (log base 10) in front of the right side and in front of
the left side

Please note that by the basic Logarithm Properties

So that we can write and using the calculator, we find that
.

As you can see, we get the same solution no matter which logarithm base we used.
However, Method 2 was a little faster because the base of the logarithm matched
the base of the exponential expression.

Problem 2:

Solve Round to 4 decimal places.

Method 1:

Let's use the natural logarithm, that is log base e.

Isolate the exponential expression

Using the Power Rule, we get

Please note that by the basic Logarithm Properties

So that we can write and using the calculator, we find that

.

Method 2:

This time we'll use the common logarithm, that is log base 10.

Isolate the exponential expression

Therefore, we have to divide both sides by

and using the calculator, we find again that .

As you can see, we get the same solution no matter which logarithm base we used.
However, in this case Method 1 was a little faster than Method 2 because the base
of the logarithm matched the base of the exponential expression.

Method 3:

This time, let's pretend that we forgot to isolate the exponential expression.

Then .

Please note that because the power x only
affects the number e and NOT the number 7 ! ! ! ! !

Instead, you MUST use the Product Rule to solve as follows

and using the calculator, we find again that .

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