In this lesson, our instructor Vincent Selhorst-Jones will give you an introduction to logarithms. Youll learn the definition of a logarithm, base 2 and then the definition of a general logarithm. Vincent also teaches shorthand notation, both in and log. This lesson also teaches you how to calculate and graph logarithms, as well as, find their domain. Lastly, youll learn about logarithms as inverse of exponentiation and get the opportunity to practice what youve learned with four full practice questions.
A logarithm is a way to reverse the process of exponentiation. It allows us to mathematically ask the question, "Given some base and some value, what exponent would we have to use on the base to create that value?"
The logarithm base a of x (loga x), where a > 0 and a ≠ 1, is defined to be the number y such that ay = x.
loga x = y ⇔ ay = x.
The idea of a logarithm can be really confusing the first few times you work with it, so make sure to watch the video to clarify how logarithms are used and what they mean.
The two most common logarithmic bases to come up are the numbers e and 10. As such, they have special notation because we have to write them so often.
The base of e is expressed as ln. It is called the natural logarithm. [Remember, e is called the natural base.]
lnx ⇔ loge x
The base of 10 is expressed with just log: if no base is given, it is assumed to be base 10. It is called the common logarithm.
logx ⇔ log10 x
Just like exponentiation, we can find the value (or a very good approximation) of a logarithm by using a calculator. Any scientific or graphing calculator will have ln and log buttons to take logarithms base e and 10, respectively. However, many calculators
will not have a way to take logarithms of arbitrary bases. There is a way around this called change of base, and we'll explore it in the next lesson, Properties of Logarithms.
When we graph logarithms, we see they grow very slowly. (This is because they are the inverse of exponentiation.) The graph of a logarithm also approaches the y-axis asymptotically. It gets very close, but it doesn't touch it.
Logarithms are the inverse of exponentiation, and vice-versa. The exponential function of base a is the inverse of the logarithmic function of base a:
f(x) = loga x f−1(x) = ax
Because a logarithm is the inverse of exponentiation, we cannot take the logarithm of some numbers. Exponentiation (ax) only has a range of (0,∞), so the corresponding logarithm can only reverse those same output values. In other words,
Domainof loga x: (0, ∞) .
We can also see the above domain must be true because it would make no sense otherwise. Consider that log2 0 = b means that 2b = 0. But no such number b exists that could do this! The same goes for negative numbers, so the domain
of any logarithm must be (0, ∞).
Note: This means we have now introduced a new type of thing to watch out for when we're looking for domains. Before, we only had to worry about dividing by 0 and having negatives under a square root. (Depending on what you've done previously, you
might also have needed to be careful about certain things with trigonometric functions.) Now we also have to watch out for taking a logarithm of 0 or a negative number. One more thing to pay attention to when you're finding the domain of a function.
Introduction to 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.