ELF - Evaluate Logarithms and Graph Logarithmic Functions Lesson

Evaluating Logarithms and Graph Logarithm Functions

From an early age, life teaches us to become accustomed to performing actions and then when given the opportunity (which happens more often than we realize) we "un-do" or reverse those actions. We do things, such as: putting a jacket on and later taking the jacket off; opening a door and closing a door; depositing money in a bank account and later withdrawing money from a bank account.

In mathematics, we perform similar actions, which are called inverse operations (these inverses "un-do" the original operation). These actions are things like adding 10 to a number and then subtracting 10 from the result; squaring a number and taking the square root of the result. We are going to study another inverse, and it is called the "logarithm function." A logarithmic function is the inverse of an exponential function, and conversely, an exponential function is the inverse of a logarithmic function.

Logarithms

Exponential and logarithmic operations undo each other since they are inverse operations.

Exponent is another word for index. The variable x is the index (exponent)
Logarithms are useful in order to solve equations in which the unknown appears in the exponent
Exponent is the logarithm
Base is always the base
y=b to the index of x is the inverse of x=log to the base times y

It is also good to see the graphs of an exponential function and logarithmic function with the same bases. The following image illustrates the inverse relationship nicely.

graph illustrating index: 2x versus log to the base of 2 times x

Next, what types of logarithms are there? There are all types of logarithms; meaning that the base number of a logarithm can be any type of number greater than "0." The most common type of logarithm is the "common logarithm," which has a base of "10."

COMMON LOGARITHMS

Common Logarithms have a base of 10. When "log x" is written, the base number of the logarithm is assumed to be "10."

Evaluating Logarithms

To evaluate logarithms, sometimes we make use of finding the inverse of a logarithmic function. In some situations, we re-write the logarithmic functions in exponential form, or we make use of the concept of "exponentiation." Exponentiation is the concept of making both sides of an equation become "exponents" with the same base of the logarithm so that the logarithm will cancel out. Whether re-writing in exponential form or using exponentiation, the goal is accomplished in that the expression can be evaluated. Here are some visual examples of re-writing logarithms in exponential form and exponentiation "at work."

Inverse Properties of Exponential and Logarithmic Functions

Logarithmic to Exponential
Given	$LaTeX: \log_{10}y=5$ $\log_{10}y=5$
Exponentiate	$LaTeX: 10^{\log10y}=10^5$ $10^{\log10y}=10^5$
"Cancels" the Logarithm	$y=10^5=100,000$

Exponential to Logarithmic
Given	$7^y=11$
Take the log of both sides with a base of "7" (performing the inverse)	$LaTeX: \log_77^y=\log_711$ $\log_77^y=\log_711$
"Cancels" the exponent (brings the variable down from the exponent)	$LaTeX: y=\log_711$ $y=\log_711$

Sometimes in evaluating logarithmic expressions and graphs, we need to enlist the "change of base formula." This formula is used because calculators only have the common log (base 10 denoted by "log") and the natural log (base e denoted by "ln"). In this lesson, we will look at the change-of-base formula using the common log, and in the next lesson we will discuss the natural log and its' applications.

Change in Base Formula

When logarithmic expressions are being evaluated (or logarithmic equations are being solved), many times the base of the logarithm is not "10." When the base number is not "10", then the change-of-base formula is applied: equation image indicator $LaTeX: \log_ab=\frac{\log\:b}{\log\:a}$ $\log_ab=\frac{\log\:b}{\log\:a}$ . The common logarithm may be used to evaluate the expression.

In evaluating the logs with another base other than 10, then we need to make use of the change-of-base formula.

Example

$LaTeX: \log_310= \\ \log_310=\frac{\log\:10}{\log\:3}\approx\frac{1}{0.4771}\approx2.096$ $\log_310= \\ \log_310=\frac{\log\:10}{\log\:3}\approx\frac{1}{0.4771}\approx2.096$

Example

$LaTeX: \log_{12}10= \\ \log_{12}10=\frac{\log\left(10\right)}{\log\left(12\right)}\approx\frac{1}{1.0792}\approx0.9266$ $\log_{12}10= \\ \log_{12}10=\frac{\log\left(10\right)}{\log\left(12\right)}\approx\frac{1}{1.0792}\approx0.9266$

Graphing Logarithmic Functions

In graphing logarithmic functions, we are going to see that they are the inverse graphs of the exponential function with the same base number. Remember from previous units that the graphs of inverse functions, are when we "switch" the domain and range of the functions. In simple terms, an exponential function with a base number of 2 has a domain of all real numbers and a range that starts close to zero (but never touches 0) and goes "up" to positive infinity. This function will have a horizontal asymptote at y = 0. Therefore, the graph of the logarithmic functions with a base of 2 will have a domain that starts close to 0 (but never touches it) and goes "right" to infinity. Its range will be all real numbers. This function will have a vertical asymptote at x = 0. In this picture below, there are three exponential function graphs and their inverse logarithmic function graphs. Again, we will discuss the exponential and logarithmic functions involving the base number "e" in our next lesson.

graphing three different logarithmic functions and the base number e

Let's start by watching the following videos that introduce us to logarithmic functions, as well as give us some examples on how to simplify, evaluate, and graph logarithmic functions.

It's now time for us to explore and practice working with Evaluating and Graphing Logarithmic Functions. Here is a visual summary (with some examples) of what we have explored.

Problem	Means	The answer is	because
$LaTeX: \log_28$ $\log_28$	2 to the what power is 8?	3	$2^3=8$
$LaTeX: \log_216$ $\log_216$	2 to the what power is 16?	4	$2^4=16$
$LaTeX: \log_210$ $\log_210$	2 to the what power is 10?	somewhere between 3 and 4	$LaTeX: 2^3=8\:and\:2^4=16$ $2^3=8\:and\:2^4=16$
$LaTeX: \log_82$ $\log_82$	8 to the what power is 2?	1/3	$LaTeX: 8^{\frac{1}{3}}=\sqrt[3]{8}=2$ $8^{\frac{1}{3}}=\sqrt[3]{8}=2$
$LaTeX: \log_{10}10,000$ $\log_{10}10,000$	10 to the what power is 10,000>	4	$10^4=10000$
$LaTeX: \log_{10}\left(\frac{1}{100}\right)$ $\log_{10}\left(\frac{1}{100}\right)$	10 to the what power is 1/100?	-2	$LaTeX: 10^{-2}=\frac{1}{10^2}=\frac{1}{100}$ $10^{-2}=\frac{1}{10^2}=\frac{1}{100}$
$LaTeX: \log_50$ $\log_50$	5 to the what power is 0?	There is no answer.	$LaTeX: 5^{something}$ $5^{something}$ will never be 0

Finding the Inverse Algebraically

Let's try to find the logarithmic function from the exponential function algebraically.

Example: Rewrite the inverse of the exponential function as a logarithmic function.	$y=2^x+3$
Beginning Equation	$y=2^x+3$
Switch the x and the y.	$x=2^y+3$
Isolate the term containing y.	$x-3=2^y$
Rewrite the exponential function as a logarithmic function	$LaTeX: y=\log_2\left(x-3\right)$ $y=\log_2\left(x-3\right)$

Image of a graph
Given a times log to the base of b(x-h)+k
The reflected function is increasing as x moves from 0 to infinite
The reflected function is decreasing as x moves from 0 to infinite
The aysmptote remains x-0
The x-intercept in both graphs is (1,0)
The Domain is (h, infinite) in these examples are (0,infinite)
The range is (negative infinite, infinite)

IMAGES CREATED BY GAVS