Exponential and Logarithmic Functions
Use the Properties of Logarithms
Learning Objectives
By the end of this section, you will be able to:
- Use the properties of logarithms
- Use the Change of Base Formula
Before you get started, take this readiness quiz.
Use the Properties of Logarithms
Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.
The first two properties derive from the definition of logarithms. Since we can convert this to logarithmic form and get Also, since we get
In the next example we could evaluate the logarithm by converting to exponential form, as we have done previously, but recognizing and then applying the properties saves time.
Evaluate using the properties of logarithms: ⓐ and ⓑ
ⓐ
ⓑ
Evaluate using the properties of logarithms: ⓐ ⓑ
ⓐ 0 ⓑ 1
Evaluate using the properties of logarithms: ⓐ ⓑ
ⓐ 0 ⓑ 1
The next two properties can also be verified by converting them from exponential form to logarithmic form, or the reverse.
The exponential equation converts to the logarithmic equation which is a true statement for positive values for x only.
The logarithmic equation converts to the exponential equation which is also a true statement.
These two properties are called inverse properties because, when we have the same base, raising to a power “undoes” the log and taking the log “undoes” raising to a power. These two properties show the composition of functions. Both ended up with the identity function which shows again that the exponential and logarithmic functions are inverse functions.
For and
In the next example, apply the inverse properties of logarithms.
Evaluate using the properties of logarithms: ⓐ and ⓑ
ⓐ
ⓑ
Evaluate using the properties of logarithms: ⓐ ⓑ
ⓐ 15 ⓑ 4
Evaluate using the properties of logarithms: ⓐ ⓑ
ⓐ 8 ⓑ 15
There are three more properties of logarithms that will be useful in our work. We know exponential functions and logarithmic function are very interrelated. Our definition of logarithm shows us that a logarithm is the exponent of the equivalent exponential. The properties of exponents have related properties for exponents.
In the Product Property of Exponents, we see that to multiply the same base, we add the exponents. The Product Property of Logarithms, tells us to take the log of a product, we add the log of the factors.
If and then,
The logarithm of a product is the sum of the logarithms.
We use this property to write the log of a product as a sum of the logs of each factor.
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible: ⓐ and ⓑ
ⓐ
ⓑ
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
ⓐⓑ
ⓐ
ⓑ
Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
ⓐⓑ
ⓐ
ⓑ
Similarly, in the