Exponential And Logarithmic Functions Essay Paper

Exponential And Logarithmic Functions Essay Paper

What is an Exponent?

2 with exponent 3 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: 23 = 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?”:

Logarithm Question

And answers it like this:

exponent to logarithm

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:

logarithm concept

(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):

Exponent vs Logarithm

They are “Inverse Functions”


Doing one, then the other, gets you back to where you started:

Doing ax then loga gives you x back again:Log a (a^x)
Doing loga then ax gives you x back again:a^(log a (x))


It is too bad they are written so differently … it makes things look strange. So it may help to think of ax as “up” and loga(x) as “down”:

going up, then down, returns you back again:down(up(x)) = x
going down, then up, returns you back again:up(down(x)) = x


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 log3(x) = 5

Start with:log3(x) = 5

We want to “undo” the log3 so we can get “x =”

Use the Exponential Function (on both sides):3^(log3(x))=3^5
And we know that 3^(log3(x))=x, so:x = 35
Answer: x = 243

And also:

Example: Calculate y in y=log4(1/4)

Start with:y = log4(1/4)
Use the Exponential Function on both sides:4^y=4^( log4(1/4) )
Simplify:4y = 1/4

Now a simple trick: 1/4 = 4−1

So:4y = 4−1
And so:y = −1

Properties of Logarithms

One of the powerful things about Logarithms is that they can turn multiply into add.

loga( m × n ) = logam + logan

“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

loga(m × n) = logam + logan the log of multiplication is the sum of the logs
loga(m/n) = logam − logan the log of division is the difference of the logs
loga(1/n) = −logan this just follows on from the previous “division” rule, because loga(1) = 0
loga(mr) = r ( logam ) the log of m with an exponent r is r times the log of m

Remember: the base “a” is always the same!

book of logarithmsHistory: 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 loga( (x2+1)4√x )

Start with:loga( (x2+1)4√x )
Use loga(mn) = logam + logan :loga( (x2+1)4 ) + loga( √x )
Use loga(mr) = r ( logam ) : 4 loga(x2+1) + loga( √x )
Also √x = x½ :4 loga(x2+1) + loga( x½ )
Use loga(mr) = r ( logam ) again: 4 loga(x2+1) + ½ loga(x)

That is as far as we can simplify it … we can’t do anything with loga(x2+1).


Answer: 4 loga(x2+1) + ½ loga(x)

Note: there is no rule for handling loga(m+n) or loga(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: loga(5) + loga(x) loga(2)

Start with:loga(5) + loga(x) − loga(2)
Use loga(mn) = logam + logan :loga(5x) − loga(2)
Use loga(m/n) = logam − logan : loga(5x/2)


Answer: loga(5x/2)

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = ax is x = ay. The logarithmic function y = logax is defined to be equivalent to the exponential equation x = ay. y = logax only under the following conditions: x = ay, a > 0, and a≠1. It is called the logarithmic function with base a.

Consider what the inverse of the exponential function means: x = ay. Given a number x and a base a, to what power y must a be raised to equal x? This unknown exponent, y, equals logax. So you see a logarithm is nothing more than an exponent. By definition, alogax = x, for every real x > 0.

Below are pictured graphs of the form y = logax 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.

Figure %: Two graphs of y = logax. On the left, y = log10x, and on the right, y = logx.

The domain of a logarithmic function is real numbers greater than zero, and the range is real numbers. The graph of y = logax is symmetrical to the graph of y = ax 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)=bx\displaystyle f{{\left({x}\right)}}={b}^{x}f(x)=bx

where b is the base and x is the exponent (or power).

If b is greater than 1\displaystyle{1}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.

calculator's index (or power) button

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 xy which is equivalent to the ^ symbol.

Example of an Exponential Function

Consider the function f(x)=2x\displaystyle f{{\left({x}\right)}}={2}^{x}f(x)=2x.

In this case, we have an exponential function with base 2\displaystyle{2}2. Some typical values for this function would be:

x −2\displaystyle-{2}2 −1\displaystyle-{1}1 0\displaystyle{0}0 1\displaystyle{1}1 2\displaystyle{2}2 3\displaystyle{3}3
f(x) 14\displaystyle\frac{1}{{4}}41 12\displaystyle\frac{1}{{2}}21 1\displaystyle{1}1 2\displaystyle{2}2 4\displaystyle{4}4 8\displaystyle{8}8

Here is the graph of y=2x\displaystyle{y}={2}^{x}y=2x.123-1-2-3-412345678-1-2xy

Open image in a new page

Graph of y=2x\displaystyle{y}={2}^{x}y=2x.Exponential And Logarithmic Functions Essay Paper


  • As x increases, y also increases.
  • As x increases, the slope of the graph also increases.
  • The curve passes through (0,1)\displaystyle{\left({0},{1}\right)}(0,1). All exponential curves of the form f(x) = bx pass through (0,1)\displaystyle{\left({0},{1}\right)}(0,1), if b>0\displaystyle{b}>{0}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.

Continues below ⇩

Logarithmic Functions

A logarithm is simply an exponent that is written in a special way.

For example, we know that the following exponential equation is true:


In this case, the base is 3\displaystyle{3}3 and the exponent is 2\displaystyle{2}2. We can write this equation in logarithm form (with identical meaning) as follows:


We say this as “the logarithm of 9\displaystyle{9}9 to the base 3\displaystyle{3}3 is 2\displaystyle{2}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)=log⁡bx\displaystyle f{{\left({x}\right)}}={{\log}_{{b}}{x}}f(x)=logbx

The base of the logarithm is b.

The 2 most common bases that we use are base 10\displaystyle{10}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=23\displaystyle{8}={2}^{3}8=23


Example 2

Write in exponential form: log⁡101000=3\displaystyle{{\log}_{{10}}{1000}}={3}log101000=3


Example 3

Find b if




1. Evaluate y=9x\displaystyle{y}={9}^{x}y=9x if x=0.5\displaystyle{x}={0.5}x=0.5


2. Express 82=64\displaystyle{8}^{2}={64}82=64 in logarithmic form.Exponential And Logarithmic Functions Essay Paper


3. Express log⁡11121=2\displaystyle{{\log}_{{11}}{121}}={2}log11121=2 in exponential form.


4. Determine the unknown: log10 0.01 = x


5. Determine the unknown, b:



Exponential And Logarithmic Functions Essay Paper


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