In this lesson, our instructor Vincent Selhorst-Jones discusses the Properties of Logarithms. First he goes over the basic properties and then inverse long(exp). Youll learn how to work with logarithms of a power, product, and quotient. Youll also learn why there is no rule for loga(M+N). Vincent ends the lesson with a summary of the properties, a change of base discussion, and then four practice problems.
We defined the idea of a logarithm in the previous lesson. A logarithm is the inverse of exponentiation:
loga x = y ⇔ ay = x.
Since logarithms and exponents are so deeply connected, we might expect logarithms to have some interesting properties, just like we discovered how exponents have many interesting properties.
If you're interested in understanding how we figure out any of the below properties, check out the video for an explanation.
From the definition of a logarithm, we can immediately find two basic properties:
loga 1 = 0 loga a = 1
Logarithms and exponentiation are inverse processes: they "cancel" each other out.
loga ax = x aloga x = x
If we take the logarithm of a power, we can "bring the power down" in front of the logarithm:
loga xn = n·loga x.
If we have the logarithm of a product, we can split it through addition of logarithms:
loga (M ·N) = loga M + loga N.
If we have the logarithm of a quotient, we can split it through subtraction of logarithms:
= loga M − loga N.
Caution! Notice that none of these properties were ever of the form loga (M+N). That's because there is just no nice formula to break apart loga (M+N).
If we have a logarithm that uses a different base than we want, we can change it through the change of base formula:
logv x =
Notice how this allows us to change from an expression logv x to an expression that only uses logu . By using u = e or u=10, we can evaluate with any calculator.
Properties of Logarithms
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.