Using the Definition of a Logarithm to Solve Logarithmic Equations
In this unit, you will explore the techniques for solving logarithmic equations. We will begin by using the definition of a logarithm to "undo" it. Then, we will work up to more complex techniques.
Using the Definition of a Logarithm to Solve Logarithmic Equations
We have already seen that every logarithmic equation is equivalent to the exponential equation . We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.
For example, consider the equation . To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for 5:
Apply the product rule of logarithms.


Distribute.


Apply the definition of a logarithm.


Add 10 to both sides.


Divide by 6.

Using the Definition of a Logarithm to Solve Logarithmic Equations
For any algebraic expression and real numbers and , where , ,
Example 9
Using Algebra to Solve a Logarithmic Equation
Solve .
Solution
Subtract 3.


Divide by 2.


Rewrite in exponential form.

Try It #9
Example 10
Using Algebra Before and After Using the Definition of the Natural Logarithm
Solve .
Solution
Divide by 2.


Divide by 6.

Try It #10
Example 11
Using a Graph to Understand the Solution to a Logarithmic Equation
Solve .
Solution
Use the definition of the natural logarithm.
Figure 3 represents the graph of the equation. On the graph, the xcoordinate of the point at which the two graphs intersect is close to 20. In other words . A calculator gives a better approximation: .
Figure 3 The graphs of and cross at the point , which is approximately (20.0855, 3).
Try It #11
Use a graphing calculator to estimate the approximate solution to the logarithmic equation to 2 decimal places.
Source: Rice University, https://openstax.org/books/collegealgebra/pages/66exponentialandlogarithmicequations
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