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}
x^{2}  8x + 16
x^{2}  2x + 16
x^{2}  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 2^{2}
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 x^{2}
x^{2}
x
x 
= log 144 = log
144
= 144
= Â±
= Â± 12 
The only solution is x =  12 since log(12) is undefined.
We leave the check to you. 