Exponential And Logarithmic Functions Essay Paper
Exponential And Logarithmic Functions Essay Paper
What is an Exponent?
The exponent of a number says how many times to use the number in a multiplication.Exponential And Logarithmic Functions Essay Paper In this example: 2^{3} = 2 × 2 × 2 = 8 (2 is used 3 times in a multiplication to get 8) |
What is a Logarithm?
A Logarithm goes the other way.
It asks the question “what exponent produced this?”:
And answers it like this:
In that example:
- The Exponent takes 2 and 3 and gives 8 (2, used 3 times in a multiplication, makes 8)
- The Logarithm takes 2 and 8 and gives 3 (2 makes 8 when used 3 times in a multiplication)
A Logarithm says how many of one number to multiply to get another number
So a logarithm actually gives you the exponent as its answer:
(Also see how Exponents, Roots and Logarithms are related.)
Working Together Exponential And Logarithmic Functions Essay Paper
Exponents and Logarithms work well together because they “undo” each other (so long as the base “a” is the same):
They are “Inverse Functions”
Doing one, then the other, gets you back to where you started:
It is too bad they are written so differently … it makes things look strange. So it may help to think of a^{x} as “up” and log_{a}(x) as “down”:
Anyway, the important thing is that:
The Logarithmic Function is “undone” by the Exponential Function.
(and vice versa)
Like in this example:
Example, what is x in log_{3}(x) = 5
We want to “undo” the log_{3} so we can get “x =”
And also:
Example: Calculate y in y=log_{4}(1/4)
Now a simple trick: 1/4 = 4^{−1}
Properties of Logarithms
One of the powerful things about Logarithms is that they can turn multiply into add.
log_{a}( m × n ) = log_{a}m + log_{a}n
“the log of multiplication is the sum of the logs”
Why is that true? See Footnote.
Using that property and the Laws of Exponents we get these useful properties:Exponential And Logarithmic Functions Essay Paper
log_{a}(m × n) = log_{a}m + log_{a}n | the log of multiplication is the sum of the logs |
log_{a}(m/n) = log_{a}m − log_{a}n | the log of division is the difference of the logs |
log_{a}(1/n) = −log_{a}n | this just follows on from the previous “division” rule, because log_{a}(1) = 0 |
log_{a}(m^{r}) = r ( log_{a}m ) | the log of m with an exponent r is r times the log of m |
Remember: the base “a” is always the same!
History: Logarithms were very useful before calculators were invented … for example, instead of multiplying two large numbers, by using logarithms you could turn it into addition (much easier!)
And there were books full of Logarithm tables to help.
Let us have some fun using the properties:
Example: Simplify log_{a}( (x^{2}+1)^{4}√x )
That is as far as we can simplify it … we can’t do anything with log_{a}(x^{2}+1).
Answer: 4 log_{a}(x^{2}+1) + ½ log_{a}(x)
Note: there is no rule for handling log_{a}(m+n) or log_{a}(m−n)
We can also apply the logarithm rules “backwards” to combine logarithms:Exponential And Logarithmic Functions Essay Paper
Example: Turn this into one logarithm: log_{a}(5) + log_{a}(x) − log_{a}(2)
Answer: log_{a}(5x/2)
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = a^{x} is x = a^{y}. The logarithmic function y = log_{a}x is defined to be equivalent to the exponential equation x = a^{y}. y = log_{a}x only under the following conditions: x = a^{y}, a > 0, and a≠1. It is called the logarithmic function with base a.
Consider what the inverse of the exponential function means: x = a^{y}. Given a number x and a base a, to what power y must a be raised to equal x? This unknown exponent, y, equals log_{a}x. So you see a logarithm is nothing more than an exponent. By definition, a^{logax} = x, for every real x > 0.
Below are pictured graphs of the form y = log_{a}x when a > 1 and when 0 < a < 1. Notice that the domain consists only of the positive real numbers, and that the function always increases as x increases.
The domain of a logarithmic function is real numbers greater than zero, and the range is real numbers. The graph of y = log_{a}x is symmetrical to the graph of y = a^{x} with respect to the line y = x. This relationship is true for any function and its inverse.
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Exponential Functions
Exponential functions have the form:
$f(x)=b_{x}$
where b is the base and x is the exponent (or power).
If b is greater than $1$, the function continuously increases in value as x increases.Exponential And Logarithmic Functions Essay Paper A special property of exponential functions is that the slope of the function also continuously increases as x increases.
It is common to write exponential functions using the carat (^), which means “raised to the power”. Computer programing uses the ^ sign, as do some calculators.
Other calculators have a button labeled x^{y} which is equivalent to the ^ symbol.
Example of an Exponential Function
Consider the function $f(x)=2_{x}$.
In this case, we have an exponential function with base $2$. Some typical values for this function would be:
x | $−2$ | $−1$ | $0$ | $1$ | $2$ | $3$ |
---|---|---|---|---|---|---|
f(x) | $41 $ | $21 $ | $1$ | $2$ | $4$ | $8$ |
Here is the graph of $y=2_{x}$.123-1-2-3-412345678-1-2xy
Graph of $y=2_{x}$.Exponential And Logarithmic Functions Essay Paper
Notice:
- As x increases, y also increases.
- As x increases, the slope of the graph also increases.
- The curve passes through $(0,1)$. All exponential curves of the form f(x) = b^{x} pass through $(0,1)$, if $b>0$.
- The curve does not pass through the x-axis. It just gets closer and closer to the x-axis as we take smaller and smaller x-values.
A logarithm is simply an exponent that is written in a special way.
For example, we know that the following exponential equation is true:
$3_{2}=9$
In this case, the base is $3$ and the exponent is $2$. We can write this equation in logarithm form (with identical meaning) as follows:
$g_{3}9=2$
We say this as “the logarithm of $9$ to the base $3$ is $2$“. What we have effectively done is to move the exponent down on to the main line. This was done historically to make multiplications and divisions easier, but logarithms are still very handy in mathematics.
The logarithmic function is defined as:
$f(x)=g_{b}x$
The base of the logarithm is b.
The 2 most common bases that we use are base $10$ and base e, which we meet in Logs to base 10 and Natural Logs (base e) in later sections.
The logarithmic function has many real-life applications, in acoustics, electronics, earthquake analysis and population prediction.
Example 1
Write in logarithm form: $8=2_{3}$
Answer
Example 2
Write in exponential form: $g_{10}1000=3$
Answer
Example 3
Find b if
$−4=g_{b}(811 )$
Answer
Exercises
1. Evaluate $y=9_{x}$ if $x=0.5$
Answer
2. Express $8_{2}=64$ in logarithmic form.Exponential And Logarithmic Functions Essay Paper
Answer
3. Express $g_{11}121=2$ in exponential form.
Answer
4. Determine the unknown: log_{10} 0.01 = x
Answer
5. Determine the unknown, b:
$g_{b}(41 )=−21 $
Answer
Exponential And Logarithmic Functions Essay Paper
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