Learning Objectives
Introduction
Tutorial
The logarithmic function with base b, where b > 0 and b 1, is denoted by
and is defined by
if and only if
This definition can work in both directions. In some cases you will have an equation written in log form and need to convert it to exponential form and vice versa.
So, when you are converting from log form to exponential form, b is your base, Y IS YOUR EXPONENT, and x is what your exponential expression is set equal to.
Note that your domain is all positive real numbers and range is all
real numbers.
First, let's figure out what the base needs to be. What do you think? It looks like the b in the definition correlates with 5 in our problem - so our base is going to be 5.
Next, let's figure out the exponent. This is very key, again remember that logs are another way to write exponents. This means the log is set equal to the exponent, so in this problem that means that the exponent has to be 3.
That leaves 125 to be what the exponential expression is set equal to.
Putting all of this into the log definition we get:
First, let's figure out what the base needs to be. What do you think? It looks like the b in the definition correlates with 7 in our problem - so our base is going to be 7.
Next, let's figure out the exponent. This is very key, again remember that logs are another way to write exponents. This means the log is set equal to the exponent, so in this problem that means that the exponent has to be y.
That leaves 49 to be what the exponential expression is set equal to.
Putting all of this into the log definition we get:
First, let's figure out what the base needs to be. What do you think? It looks like the b in the definition correlates with 6 in our problem - so our base is going to be 6.
Next, let's figure out the exponent. In this direction it is easy to note what the exponent is because we are more used to it written in this form, but when we write it in the log form we have to be careful to place it correctly. Looks like the exponent is -2, don't you agree?
The value that the exponential expression is set equal to is what goes inside the log function. In this problem that is 1/36.
Let's see what we get when we put this in log form:
Rewriting the original problem using exponents we get:
First, let's figure out what the base needs to be. What do you think? It looks like the b in the definition correlates with 81 in our problem - so our base is going to be 81.
Next, let's figure out the exponent. In this direction it is easy to note what the exponent is because we are more used to it written in this form, but when we write it in the log form we have to be careful to place it correctly. Looks like the exponent is 1/2, don't you agree?
The value that the exponential expression is set equal to is what goes inside the log function. In this problem that is x.
Let's see what we get when we put this in log form:
Let's step through a few examples of this:
*Rewriting in exponential form
*x is the exponent
we need on 4 to get 64
*Rewriting in exponential form
*x is the exponent
we need on 9 to get 1
*Rewriting in exponential form
*x is the exponent
we need on 7 to get 7
*Rewriting in exponential form
*x is the exponent
we need on 5 to get square root of 5
Looks like the base is 3, the exponent is y,
and the log will be set = to x:
The first two columns just show what values we are going to plug in for y.
The last three columns show the corresponding values for x and y for the given function.
x
y
y
(x, y)
-2
-2
(1/9, -2)
-1
-1
(1/3, -1)
0
0
(1, 0)
1
1
(3, 1)
2
2
(9, 2)
Looks like the base is 3, the exponent is y,
and the log will be set equal to x + 1:
The first two columns just show what values we are going to plug in for y.
The last three columns show the corresponding values for x and y for the given function.
x
y
y
(x, y)
-2
-2
(-8/9, -2)
-1
-1
(-2/3, -1)
0
0
(0, 0)
1
1
(2, 1)
2
2
(8, 2)
Next, we need to write in exponential form, just like we practiced in examples 1 and 2.
Looks like the base is 3, the exponent is -y,
and the log will be set equal to x:
The first two columns just show what values we are going to plug in for y.
The last three columns show the corresponding values for x and y for the given function.
x
y
y
(x, y)
-2
-2
(9, -2)
-1
-1
(3, -1)
0
0
(1, 0)
1
1
(1/3, 1)
2
2
(1/9, 2)
Since x is part of the inside of the log
on this problem we need to find a value of x,
such that the inside of the log, 5 - x, is
positive.
*Domain of this function
That means that if we put in any value of x that is less than 5, we will end up with a positive value inside our log.
Note how on this problem the inside of the log is squared. So
no matter what we plug in for x, the inside
will always be positive or zero. Since we can only have positive
values inside the log, our only restriction is where the inside would
be 0.
,
where b > 0 and b is not equal to 1.
Boy, the definition of logs sure does come in handy to explain these properties - applying that definition you would have b raised to the r power which equals b raised to the r power.
Here is a quick illustration of how this property works:
,
where b > 0 and b is not equal to 1.
This one is a little bit more involved and weird looking huh?
Going back to our favorite saying - a log is another way to write exponents
- what we have here is the log is the exponent we need to raise b to get m, well if we turn around an raise our
first base b to that exponent, it stands to
reason that we would get m.
Here is a quick illustration of how this property works:
When using common log (base 10), use the form log x to write it.
Natural Log
When using the natural log (base e),
use the form ln x to write it.
I'm going to use the first inverse property shown above:
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1b: Express the given logarithmic equation exponentially.
Practice Problems 2a - 2b: Express the given exponential equation in a logarithmic form.
Practice Problems 3a - 3d: Evaluate the given log function without using a calculator.
Practice Problems 4a - 4b: Graph the given function.
Practice Problem 5a: Find the domain of the given logarithmic function.
Practice Problems 6a - 6b: Evaluate the given expression without the use of a calculator.
Practice Problems 7a - 7b: Simplify the given expression without the use of a calculator.
Need Extra Help on these Topics?
The following are webpages that can assist you in the topics that were covered on this page.
http://www.purplemath.com/modules/graphlog.htm
This webpage helps you with graphing log functions.
http://www.sosmath.com/algebra/logs/log4/log4.html#logarithm
This webpage helps you with the definition of and graphing logs.
Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.
Last revised on March 22, 2011 by Kim Seward.
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