is one in which a
variable occurs in the
exponential equation in
which each side can be
terms of the
same base can
be solved using the
If the bases are the
same, set the exponents
Since the bases are the same, set
the exponents equal to one another:
2x + 1 = 3x - 2
3 = x
27 can be expressed as a power of
2x - 1 = 3x -1 = x
25 can be expressed as a power of
3x - 8 = 4x
-8 = x
If you can express both sides of the equation as
powers of the samebase, you can
set the exponents equal to solve for x.
Unfortunately, not all exponential
equations can be expressed in terms of a common base.
For these equations, logarithms are used to arrive at a
solution. (You may solve using common log or
To solve most
1. Isolate the
2. Take log or ln of both sides.
3. Solve for the variable.
Things to remember
Solve for x, to the nearest thousandth.
• Take the log of both
• Apply the log rule for exponents shown
• Solve for x.
• Estimate answer using calculator.
• log base 5 can also be used as a solution
• notice how the log5 of 5xis reallycomposition and yields
• notice the change of base formula used
Also, log base 3 can be used.
Also, log base 1/2 can be used as a
Since the natural log is the inverse
of the exponential function, use ln to
quickly solve this problem.
• First, get
rid of the coefficient of the exponential
term (divide by 150).
• Now, proceed using ln to quickly
• Do not round too quickly. Be sure
to carry enough decimal values to allow you
to round to thousandths (in this case) for the final
• Isolate the exponential
• Take the log of both sides
• Apply the log property
• Divide by log 4
• Estimate using calculator
• Isolate the
• Divide each side by
the coefficient of 2
• Take ln of both sides
• Remember that ln x and ex
are inverse functions.
• First, divide by the
coefficient to isolate the exponential
Proceed as shown above.
This question requires some additional thinking.
Because of the differing powers of e, our
previous methods will not be of much help. We
will need a different strategy with this problem.
• Remember that ex • ex
problem is really x2 - 4x + 3 = 0
where x = ex.
• Both solutions are answers.
How to use your
TI-83+/84+ graphing calculator with
exponential equations. Click calculator.