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Solving Logarithmic Equations using Exponents

Solving Logarithmic Equations using Exponents

Author: Sophia Tutorial
Description:

Solve a logarithmic equation by rewriting into an exponential equation.

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Tutorial
what's covered
  1. Exponential-Logarithmic Relationships
  2. Rewriting Logarithmic Equations as Exponential Equations
  3. Using the Log Expression as an Exponent

1. Exponential-Logarithmic Relationships

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

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

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. For example, 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
Logarithmic equation
7 to the power of 2.8 end exponent equals x
Rewrite into an exponential equation
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

4 log subscript 5 open parentheses x close parentheses equals 7.6
Logarithmic equation
log subscript 5 open parentheses x close parentheses equals 1.9
Divide by 4 to have only log subscript 5 open parentheses x close parentheses on left side
5 to the power of 1.9 end exponent equals x
Rewrite into an exponential equation
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 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 that you may be used to. We can use the base of the logarithm as a base to 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.

log subscript 3 open parentheses 6 x plus 9 close parentheses equals 4
Logarithmic equation
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
Use 3 as a base
6 x plus 9 equals 81
Simplify both sides
6 x equals 72
Subtract 9
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.