Math 20 Cohort Solving Logarithmic Equations Activity
Properties of Logarithms
One of the most important skills we learn in Intermediate Algebra is how to solve Logarithmic Equations. In this review activity our goal is to review how to solve some of the most popular types as well as review strategies and definitions.
Type 1) Logarithmic equations that can be solved by appealing to the definition and using the counter clockwise method of extracting a solution.
If the logarithmic function with base b is defined by
Where the domain of the logarithmic function is the interval. The range is the interval
Example: Given Find the value of x.
Using the definition of Log and counter clockwise method
Next we square both sides of the equation.
Solve the following using the counter clockwise method.
1) 4)
2) 5)
3) 6)
Type 2) Logarithmic Equations that can be solve by appealing the following theorem:
(Same base, we may equate arguments)
Recall:
Common Logarithms: Let y and x be real numbers, with x > 0.Then
Natural Logarithms: Let y and x be real numbers, with x > 0.Then
Example: Solve for x.
To solve we use properties of logarithms to write each side of the equation as a logarithm of a single quantity, then use the above theorem and solve.
By Theorem
7)
8)
9) (are both answers valid, if so why?)
10)
11) (Remember same rules apply)
12)
Type 3) The next type is when we have equations that contain terms involving logarithms and constants.
The strategy here is to place all the terms involving logarithms on one side of the equations and the constants on the other. Then write the side containing logarithms into a single quantity. Finally use the counter clockwise method to get the variable out of the log function and solve.
Example: Solve and check the following:
Therefore by the Zero Product Principle x = 5 or x = -2 (extraneous)
13)
14)
Let us conclude our activity by reviewing “The Change of Base Property”. The change of base property
Is used to write a logarithm in terms of quantities that can be evaluated with a calculator so that we may check our answer or obtain approximate values. The bases that our calculators can evaluate for us are
the “Common Log (Base 10) ” and the “Natural Log ( Base e ) ”.
Example: Write the following using both change of base formulas.
Therefore
Write the following using both change of base formulas.