LOGARITHMIC EQUATIONS
Just like with exponential equations, there are basically two types of logarithmic equations.
(1) Logarithmic equations in which two logs can be compared or
(2) Logarithmic equations in which the logs cannot be compared.
(1) Logarithmic equations in which two logs can be compared
Given a logarithmic equation containing two logarithms and NO CONSTANTS you can usually use the
following property:
If log a M = log a N , then M = N
Example:
Solve log 3 (2 x − 4) = 2 log 3 4
Solution:
Notice that there are two logarithms in this equation and NO CONSTANTS. This is a good
indication that we can use the property listed above. In order to use this property, use another log
property to simplify the right hand side of the equation:
log 3 (2 x − 4) = log 3 42
log 3 (2 x − 4) = log 3 16
Use the property log a M r = r log a M
Simplify.
If log a M = log a N , then M = N
2 x − 4 = 16
x = 10
Solve.
Make sure to check for extraneous solutions. In this example, if you plug x = 10 back into the
original equation, the solution checks!
(2) Logarithmic equations in which the logs cannot be compared.
Example:
Solution:
Solve log 4 ( x − 1) − log 4 (2 x + 2) = 3
First notice that there is a constant 3 in this problem. Therefore, we cannot use the technique
used in the last example. At this point, your thought process should be to first combine all
the logs using log properties, then rewrite the log as an exponent!
log 4 ( x − 1) − log 4 (2 x + 2) = 3
 x −1 
log 4 
=3
 2x + 2 
x −1
43 =
2x + 2
x −1
64 =
2x + 2
64(2 x + 2) = x − 1
128 x + 128 = x − 1
127 x = -129,
x=
M 

N
Use the property log a M − log a N = log a 
Rewrite as an exponent.
Simplify.
-129
127
Multiply both sides of the equation by 2 x + 2 .
Distribute.
Solve for x (Note that this solution DOES NOT check, since x is negative!)
EXPONENTIAL EQUATIONS
There are basically two types of exponential equations:
(1) Exponential equations in which the bases of the exponents can be made the same or
(2) Exponential equations in which the bases of the exponents cannot be made the same.
(1) Exponential equations in which the bases of the exponents can be made the same
Always first try to see if you can relate the bases. If you can relate the bases then we can use the following
property:
If a u = a v , then u = v
1
81
Example:
Solve 3x−1 =
Solution:
Notice that we can “relate the bases” on each side of the equation since 81 is a power of 3.
1
34
x−1
3 = 3−4
3x−1 =
Therefore, we can use the property above to say that x − 1 = −4, so x = −3.
(2) Exponential equations in which the bases cannot be made the same
If you cannot relate the bases you need to use a logarithm to solve.
Example:
Solve 3x −1 = 5 x
Solution:
First notice that you cannot write 3 as a power of 5 or vice versa. So we must use a logarithm in
order to “bring down” the exponents.
3x −1 = 5 x
ln 3x −1 = ln 5 x
( x − 1) ln 3 = x ln 5
x ln 3 − ln 3 = x ln 5
x ln 3 − x ln 5 = ln 3
x(ln 3 − ln 5) = ln 3
ln 3
x=
ln 3 − ln 5
Take the log of both sides (it does not matter what base you use!)
Use the property log a M r = r log a M .
Distribute.
Isolate the variables on one side.
Factor. (Students often overlook this crucial step)
Divide both sides by ln 3 − ln 5 .