Solving Equations with Log Terms and Other
Terms
Hereâ€™s how to solve an equation that contains more than one log and a
constant term.
Procedure â€”
To Solve an Equation With Log Terms and Other Terms
Step 1 Rewrite the equation with the logs on one side and the
constant term on the other side.
Step 2 Combine the logs into a single log.
Step 3 Convert the equation to exponential form and solve.
Example 1
Solve: log_{6} x + log_{6} (x  5) = 2
Solution
Step 1 
Rewrite the equation with the logs
on one side and the constant term
on the other side.
The logs are already on one
side of the equation.

log_{6} x + log_{6}(x  5) = 2 
Step 2 
Combine the logs into a
single log.
Use the Log of a Product Property.

log_{6} x(x  5)
= 2 
Step 3 
Convert the equation to
exponential form and solve.
Convert to exponential form.
Simplify each side.
Subtract 36 from both sides.
Factor.

6^{2} = x(x  5)
36 = x^{2}  5x
0 = x^{2}  5x  36
0 = (x  9)(x + 4) 

Use the Zero Product Property.
Solve for x. 
x  9 = 0
x = 9 
or
or 
x + 4
x 
= 0 = 4 
log6 x is defined only when x is positive. Therefore, x = 4 is not a
solution.
Thus, x = 9 is the solution of log_{6} x + log_{6}(x  5)
= 2. We leave the
check to you.
Example 2
Solve: ln 20x^{2} = 4.73 + ln 5x.
Step 1 
Rewrite the equation with the logs
on one side and the constant term
on the other side.
Subtract ln 5x from both sides. 
ln 20x^{2}
= 4.73 + ln 5x 
Step 2 
Combine the logs into a
single log.



Use the Log of a Quotient Property. 


Simplify the argument of the log.
Write ln 4x as log_{e} 4x. 
ln 4x = 4.73 log_{e} 4x
= 4.73 
Step 3 
Convert the equation to exponential
form and solve.
Convert to exponential form. 
e^{4.73} = 4x 

Divide both sides by 4. 


Find e^{4.73} and leave the result on the
calculator display. Then, divide by 4
and round. 
28.32
≈ x 
So, x
≈ 28.32. We leave the check to you.
