Exponential equations generally have the unknown variable as the power. The
following are examples of exponential equations:
You should already be familiar with exponential notation. Solving exponential
equations is simple, if we remember how to apply the laws of exponentials.
Investigation : solving exponential equations
Solve the following
equations by completing the table:
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
Algebraic solution
Equality for Exponential Functions
If
is a positive number such
that
, (except when
) then:
if and only if:
(If
, then
and
can differ)
This means that if we can write all terms in an equation with the same base, we
can solve the exponential equations by equating the indices. For example takethe equation
. This can be written as:
Since the bases are equal (to 3), we know that the exponents must also be equal.
Therefore we can write:
This gives:
Method: solving exponential equations
Try to write all terms with the same base.
Equate the exponents of the bases of the left and right hand side of the equation.
Solve the equation obtained in the previous step.
Check the solutions
Investigation : exponential numbers
Write the following with the same
base. The base is the first in the list. For example, in the list 2, 4, 8, thebase is two and we can write 4 as
.
2,4,8,16,32,64,128,512,1024
3,9,27,81,243
5,25,125,625
13,169
,
,
,
Solve for
:
All terms are written with the same base.
Since both sides are equal, the answer is correct.
is the solution to
.
Solve:
Since both sides are equal, the answer is correct.
is the solution to
.
Solving exponential equations
Solve the following exponential equations:
Solve:
Solve for
:
The growth of an algae in a pond can be modeled by the function
. Find the value of
such that