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 log_b(x)=y is equivalent to the exponential equation b^y=x. 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 log_2(2)+log_2(3x−5)=3.  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 5x:

Apply the product rule of logarithms.
Apply the definition of a logarithm.
Calculate 2^3.
Add 10 to both sides.
Divide by 6.

Using the Definition of a Logarithm to Solve Logarithmic Equations

For any algebraic expression S and real numbers b and c, where b > 0 , b≠1,

log_b(S)=c if and only if b^c=S

Example 9

Using Algebra to Solve a Logarithmic Equation

Solve 2lnx+3=7.


Subtract 3.
Divide by 2.
Rewrite in exponential form.

Try It #9

Solve 6+lnx=10.

Example 10

Using Algebra Before and After Using the Definition of the Natural Logarithm

Solve 2ln(6x)=7.


In (6x) = \dfrac{7}{2}
Divide by 2.
6x = e^{(\dfrac{7}{2})}
Use the definition of ln.
x = \dfrac{1}{6} e^{(\dfrac{7}{2})}
Divide by 6.

Try It #10

Solve 2ln(x+1)=10.

Example 11

Using a Graph to Understand the Solution to a Logarithmic Equation

Solve ln x=3.


ln x=3

x=e^3  Use the definition of the natural logarithm.

Figure 3 represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20. In other words e^3≈20. A calculator gives a better approximation: e^3≈20.0855.

Graph of two questions, y=3 and y=ln(x), which intersect at the point (e^3, 3) which is approximately (20.0855, 3).

Figure 3 The graphs of y=ln x and y=3 cross at the point (e^3,3), which is approximately (20.0855, 3).

Try It #11

Use a graphing calculator to estimate the approximate solution to the logarithmic equation 2^x=1000 to 2 decimal places.

Source: Rice University,
Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 License.