Use Sophia to knock out your gen-ed requirements quickly and affordably. Learn more
×

Solving Logarithmic Equations using Exponents

Author: Sophia

what's covered
In this lesson, you will learn how to solve a logarithmic equation by rewriting it into an exponential equation. Specifically, this lesson will cover:

Table of Contents

1. Exponential-Logarithmic Relationships

Logarithmic equations can be equivalently written using exponents. In general, we can say that the following two equations are equivalent:

Exponential Equation Logarithmic Equation
y equals b to the power of x log subscript b open parentheses y close parentheses equals x

Notice that the base of the logarithm is the base of the exponential expression. Additionally, the input of the logarithmic function is the output of the exponential function.


2. Rewriting Logarithmic Equations as Exponential Equations

For some logarithmic equations, it may be helpful to rewrite the equation as an equivalent exponential equation to solve.

EXAMPLE

Solve the logarithmic equation log subscript 7 open parentheses x close parentheses equals 2.8.

log subscript 7 open parentheses x close parentheses equals 2.8 Rewrite into an exponential equation
7 to the power of 2.8 end exponent equals x Evaluate using calculator
232.42 equals x Our solution

big idea
One strategy in solving logarithmic equations is to rewrite it as an exponential equation. In many cases, by doing so, the equation will have an exponential expression on one side of the equation, and an isolated variable on the other side. We can then evaluate the exponential expression to find the solution to the equation.

EXAMPLE

Solve the logarithmic equation 4 log subscript 5 open parentheses x close parentheses equals 7.6.

4 log subscript 5 open parentheses x close parentheses equals 7.6 Divide both sides by 4 to have only log subscript 5 open parentheses x close parentheses on left side
log subscript 5 open parentheses x close parentheses equals 1.9 Rewrite into an exponential equation
5 to the power of 1.9 end exponent equals x Evaluate using calculator
21.28 Our solution

hint
It is important to the log expression by itself on one side. In the example above, we had to divide both sides by 4 before we could convert from a logarithmic equation to an exponential equation.


3. Using the Log Expression as an Exponent

Another method to solving log equations involves applying the inverse relationship between exponents and logs in a slightly different way than you may be used to. We can use the base of the logarithm as a base of an exponent and place the logarithmic expression as an exponent in the equation. We'll have to do this to both sides of the equation.

EXAMPLE

Solve the logarithmic equation log subscript 3 open parentheses 6 x plus 9 close parentheses equals 4.

log subscript 3 open parentheses 6 x plus 9 close parentheses equals 4 Apply the base of the log, 3, as a base of an exponent on both sides.
3 to the power of log subscript 3 open parentheses 6 x plus 9 close parentheses end exponent equals 3 to the power of 4 On the left side, the argument of the log 6 x plus 9 remains; on the right side, evaluate 3 to the power of 4
6 x plus 9 equals 81 Subtract 9 from both sides
6 x equals 72 Divide both sides by 6
x equals 12 Our solution

big idea
This is essentially a more explicit explanation of the relationship between logarithmic equations and exponential equations. We can use the base of the log to create an exponential equation with the same base. Since logarithms and exponents are inverse operations, this undoes any log or exponent operation, leaving only the argument of the log on one side of the equation, and an exponential expression on the other.

summary
The exponential-logarithmic relationship says that we can use properties of logarithms to solve logarithmic equations, such as the power property, change of base formula, and the conversion between logarithmic form and exponential form. One method of solving logarithmic equations involves rewriting the logarithmic equation to an exponential equation. A second method involves using the log expression as an exponent, which cancels out the logarithm operation, and solving the resulting equation.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License