Logarithmic Equations
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Remember that the composition of a function and its inverse returns the starting value.

A logarithmic equation can be solved using the properties of logarithms along with the use of a common base.

Properties of Logs:




To solve most logarithmic equations:
1.  Isolate the logarithmic expression.
(you may need to use the properties
to create one logarithmic term)
2.  Rewrite in exponential form
(with a common base)
3.  Solve for the variable.

Things to remember about logs:

Do you see composition of a function and its inverse at work in the last statements?  Find out more about
exponential and log functions.

Examples:

  Solve for x: Answer:
1.
ANSWER:
    


Isolate the log expression

Choose base 10 to correspond with log (base 10)

Apply composition of inverses and solve.

2.
ANSWER:
    

Remember that ex and ln x are inverse functions.
3.
ANSWER:
    
 
4.  
ANSWER:
    

Isolate the logarithmic expression first.
5.
 
ANSWER:
    


Use the log property to express the two terms on the left as a single term.

Remember that log of a negative value is not a real number and is not considered a solution.

6.
ANSWER:    
7. Using your graphing calculator, solve for x to the nearest hundredth.
                     


 
Method 2: 
Place the left side of the equation into Y1 and the right side into Y2.  Under the CALC menu, use #5 Intersect to find where the two graphs intersect.    
            

ANSWER:   
Method 1:  Rewrite so the equation equals zero.

Find the zeros of the function.


Both values are solutions, since both values allow for the ln of a positive value.

 

How to use your
TI-83+/84+ graphing calculator  with logarithms.
Click calculator.