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Logarithmic Functions

Did you ever notice that seemingly everything in math can be graphed? Well, that goes for logarithms as well. The basic logarithmic function is $y={\mathrm{log}}_{b}\left(x\right)$ , where x and b are both greater than zero and $b\ne 1$ . Here is what the graph of a logarithmic function looks like using the common logarithm, $y={\mathrm{log}}_{}\left(x\right)$ , as an illustrative example. Recall that we assume a base of 10 whenever a logarithm lacks a subscript:

In this article, we'll explore the properties of logarithmic functions and how we can modify the graph above by changing the function (or how the function affects the graph). We'll also look at how the function rules we know already apply to logarithmic functions. Let's get started!

The domain of all logarithmic functions is the set of positive, real numbers. Something like $f\left(0.001\right)=\mathrm{log}\left(0.001\right)={10}^{-3}$ might sound like negative numbers would work for the domain, but remember what you're really saying there is $\frac{1}{{10}^{-3}}$ : a ratio between two positive numbers that will always yield a positive number. Similarly, you cannot raise any number to a power and get zero as the result, so logarithmic functions are undefined for zero as well.

However, the range of logarithmic functions encompasses all real numbers including zero and negative numbers because there are no limitations on what power we can raise numbers to. Note that the domain and range of logarithmic functions are the exact opposite of exponential functions such as $g\left(x\right)={b}^{x}$ where the domain is all real numbers but the range excludes negative numbers and zero. This makes sense as logarithmic functions are the inverse of exponential functions.

Logarithmic functions are also continuous, meaning that there are no breaks in the line. When we're drawing them, we have to remember to include the arrows on either side to indicate that it stretches infinitely in either direction. Logarithmic functions are also one-to-one, meaning that there is exactly one y value for every x value.

Finally, the y-axis is the asymptote of the graph of any basic logarithmic function. The graph of the common logarithmic function also has an x-intercept of 1 since it crosses the point $\left(1,0\right)$ .

Shifting logarithmic functions

While the base function $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ has the general shape indicated above, changing the base or the value in parenthesis can shift the image. More specifically, we can shift the image k units vertically and h units horizontally with the following equation:

$y={\mathrm{log}}_{b}\left(x+h\right)+k$

For vertical shifts, the graph shifts k units up if $k>0$ and k units down if $k<0$ .

For horizontal shifts, the graph shifts h units left if $h>0$ and h units right if $h<0$ .

Let's take a look at a practice question to see how these shifts work:

Graph the function $y={\mathrm{log}}_{}\left(x-1\right)+2$

We need to graph the base function $f\left(x\right)={\mathrm{log}}_{}\left(x\right)$ first, so go ahead and copy the image above. Next, look at how our function differs from the one we just graphed. We have an h value of -1, so we're shifting it one unit to the right. We also have a k value of 2, so we're shifting it 2 units upward. Plot the new graph on the same Cartesian plane you graphed the first one on, giving us an answer that should look something like this:

One of the hardest parts of this is remembering the difference between k and h. Since the h value controls horizontal movement, you can focus on the word "horizontal" starting with the letter h. Similarly, h precedes k in the alphabet, so h comes first in the formula above.

The natural logarithmic function

As you know, the common logarithm and natural logarithm are the most frequently seen logarithms. Natural logarithms are typically denoted using ln(x) and have a base of e: a constant sometimes called the most important number in calculus. We graphed the common logarithm above, so you might be wondering what the natural logarithm looks like when graphed. Here it is:

The overall shape is the same, which makes sense since both are logarithmic functions. That said, the y values increase a little more quickly above than they do with the graph of the common logarithm. The natural logarithm function is also the inverse of the natural base exponential function, $y={e}^{x}$ .

Logarithmic functions practice questions

a. In your own words, explain why logarithmic functions cannot have a domain of zero or a negative number.

Logarithms ask the question "What power would I need to raise this to in order to get a specific answer?" a ratio of positive numbers will always produce a positive number. Similarly, no number to any power is equivalent to zero. Therefore, logarithmic functions are undefined for negative numbers and zero

b. Find the y-intercept of the following function to two decimal places: $f\left(x\right)=\mathrm{ln}\left(x+4\right)-7$

This function involves the natural logarithm, but we find the y-intercept the same way we would for any other function: set x to 0 and solve. The equation is:

$y=\mathrm{ln}\left(x+4\right)-7$

Subbing in 0 for x, we have:

$y=\mathrm{ln}\left(0+4\right)-7$

$y=\mathrm{ln}\left(4\right)-7$

$y\approx 1.39-7$

$y\approx -5.61$

The question asks us for the y-intercept, so the answer must be an ordered pair: $\left(0,-5.61\right)$ .

c. Consider the following diagram:

What is the base of the logarithmic function pictured above?

Any base with an exponent of 1 is equal to the base, so we need to find the x value when $y=1$ . Looking at the graph, it crosses the point $\left(0.5,1\right)$ , meaning that the base we're looking for is 0.5.

d. Using graph paper, sketch a graph of $f\left(x\right)={\mathrm{log}}_{5}\left(x\right)$

The function $f\left(x\right)={\mathrm{log}}_{5}\left(x\right)$ calls for raising 5 to the y power, with x being the resulting product. The easiest way to plot it is to work out two points and go from there. For instance, x will be 5 when $y=1$ because ${5}^{1}=5$ . Therefore, one point is $\left(5,1\right)$ . Likewise, $\left(125,3\right)$ will be a point because ${5}^{3}=125$ . Now that we have two points, connect them while keeping the shape of a logarithmic function in mind to sketch your graph. It should look something like this:

Flashcards covering the Logarithmic Functions

Algebra II Flashcards

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