Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.7 Exponential Functions
Forms
A Logarithmic Function is really only formatted in one way: f(x)= logb(x). This is read as “log base b of x”. Recall that a logarithm is just a different way of phrasing an exponent. It is the inverse of the exponential so logb(x)=y is the same as by=x.
For that reason, the base is restricted in the same way that an exponential functions base is. b>0 and b≠1.
Graph
Using the knowledge that the logarithm and exponential are inverse functions we can graph a logarithm be mirroring the exponential over the line y=x.
Example 1: Graph a sketch of f(x)= -3log2(x-10)
Solution: In terms of transformations, this is a logarithmic graph flipped vertically, vertically stretched by a factor of 3 and shifted to the right 10 units:
Domain
The domain of a logarithmic function is restricted because the base is restricted. If the base can only be positive numbers, the input must be only positive numbers. This is because when raising a positive number, the base, to an exponent, y, the only results are positive numbers, x.
So x∈(0,∞).
Example 2: Find the domain of f(x)= -3log2(x-10)
Solution: In this case, the domain is restricted to when (x-10) is positive. So x∈(10,∞).
Range
The output is representative of the possible exponents, so there are no restrictions here. Any base could be raised to any numerical exponent and have an answer. So y∈R.
Special Forms
There are two common forms of logarithms. The first of these is y=log(x). This form does not have a base. What that really means is that the base is 10, because we use a base-10 number system. This is the standard logarithm and it evaluates the magnitude of the input.
The other common form is the natural logarithm which is denoted y=ln(x). This is simply a special case were the base is the special number e. That is ln(x)= loge(x).
A formula that demonstrates all of the possible transformations could be written as:
y=a logb(cx+d)+e
Applications
Logarithms are most helpful for evaluating or comparing relationships with large differences in value. A few common examples are the pH scale in Chemistry, the loudness of sound, the energy released by an earthquake, the brightness of a star in astronomy, solving exponential equations, and more. Revisiting some applications from exponential equations allows us to make new conclusions.
In Exponential Decay, logarithms can be used to calculate half-lives of radioactive isotopes.
Example 3: Find the half-life of the isotope Radon-222 if an initial sample has 1 gram of Radon, but after 8.881 days only 0.2 grams of Radon-222 remain.
Solution: Using the equation P=P0bx, first solve for b. Then, set up the equation to solve for the half-life which is when the output is half of the original value.
Review
A Logarithmic Function is the Inverse of an Exponential Function.
The base, b, should be positive.
Domain: D: x>0.
Range: R: y∈R
Equations
