Solving Equations with Log Terms and Other
Terms
Example
Solve: log_{5} (5x + 10)  1 = log_{5} 25.
Solution 

log_{5} (5x + 10)  1 = log_{5}
25 
Step 1 
Rewrite the equation with the logs
on one side and the constant term
on the other side.
To make the calculations easier,
rewrite the equation so the constant
term is positive.
Add 1 to both sides.
Subtract log_{5}25 from both sides. 
log_{5} (5x + 10) = log_{5}
25 + 1 log_{5} (5x + 10)
 log_{5} 25 = 1 
Step 2 
Combine the logs into a
single log.



Use the Log of a Quotient Property. 

Step 3 
Convert the equation to exponential
form and solve.



Convert to exponential form. Factor numerator and denominator.
Cancel the common factor of 5.



Multiply both sides by the LCD, 5.
Subtract 2 from both sides. 
25 = x + 2
23 = x 
So, the solution is x = 23.
Note:
Here is a check of the solution:
Is Is
Is
Is
Is
Is 
log_{5} (5x + 10) 
1 log_{5} (5
Â· 23 + 10)  1
log_{5} (125)  1
log_{5} (5^{3})  1
3  1
2 
= log_{5} 25
= log_{5} 25
= log_{5} 25
= log_{5} 5^{2}
= 2
= 2 
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? Yes 
