Solving Equations with Log Terms on Each Side
Example 1
Solve: 2ln(x - 4) = ln(40 - 6x)
| Solution
Use the Log of a Power Property.
Use the Principal of Logarithmic Equality.
Square the left side.
Add 6x to both sides.
Subtract 40 from both sides.
Factor.
|
2ln(x - 4)
ln(x - 4)2
(x - 4)2
x2 - 8x + 16
x2 - 2x + 16
x2 - 2x - 24
(x + 4)(x - 6) |
= ln(40 -
6x)
= ln(40 - 6x)
= 40 - 6x
= 40 - 6x
= 40
= 0
= 0 |
|
Use the Zero Product Property.
Solve for x. |
x + 4
x |
= 0
= -4 |
or
or |
x - 6 = 0
x = 6 |
|
It appears that there are two solutions. However, we can only take the log
of a positive number. Therefore, we must check each answer to be sure
that it leads to taking the log of a positive number.
| Check the first solution, x = -4:
Substitute -4 for x.
Simplify. Since ln(-8) is undefined, x
= -4 is not a solution. |
|
2ln(x - 4)
2ln(-4 - 4)
2ln(-8) |
= ln(40 -
6x)
= ln(40 - 6(-4)) ?
= ln(64) ? |
| Check the second solution, x = 6:
Substitute 6 for x.
Simplify.
Use the Log of a Power Property.
Simplify. |
Is
Is
Is
Is |
2ln(x - 4) 2ln(6
- 4)
2ln 2
ln 22
ln 4 |
= ln(40 -
6x)
= ln(40 - 6(6)) ?
= ln 4 ?
= ln 4 ?
= ln 4 ? Yes |
So, x = 6 is the solution.
Example 2
Solve: 2log x = log 144
| Solution
Use the Log of a Power Property.
Use the Principal of Logarithmic Equality.
Take the square root of both sides.
Simplify. |
2log x
log x2
x2
x
x |
= log 144 = log
144
= 144
= ±
= ± 12 |
The only solution is x = - 12 since log(-12) is undefined.
We leave the check to you. |
