In the series: Note 19.

Version: 0.5

By: Albert van der Sel

Doc. Number: Note 19.

For who: for beginners.

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Status: Ready

Maybe you need to pick up "some" basic "mathematics" rather

So really..., my emphasis is on "rather

So, I am not sure of it, but I hope that this note can be of use.

Ofcourse, I hope you like my "style" and try the note anyway.

Each note in this series, is build "on top" of the preceding ones.

Please be sure that you are on a "level" at least equivalent to the contents up to, and including, note 18.

At least, before you start here, take a look at note 7, the "b

1. Very Quick recap exponential functions y=a

2. Logarithmic functions.

3. A very quick recap on working with Powers.

4. Some important properties of Logarithms.

5. Solving equations with exponents and logarithms.

6. The domain for logarithmic functions (for which "x" are they valid?).

In figure 1, we see 4 examples of the function f(x)=a

Take a look at f(x)=2

in that expression, to find the corresponding "y" values.

Let's try a few: 2

You can see that the values found, match the graph. Further, it's a rising function, if x increases.

For the fourth figure, the base number is "0.3". You can perform the same sort of excercise, and find out

that it is a decreasing function. For example, for x=2, we have 0.3

So, if x=4, we would have 0.3

But it never actually will be "0", since the line y=0, is it's asymptote.

So, in general, for f(x)=a

- If a>1, it's an ascending function.

- And for a in [0,1], it's a decending function.

But the "borders" 0 and 1, are usually left out. Why?

Well, f(x)=0

We do not

-Now take a look at f(x)=2

added to f(x)=2

compared to f(x)=2

Note, that in this case, the asymptote is the line "y=3".

-If we would have f(x)=2

Note, that in that case, the asymptote is the line "y = -3".

-Now, what if a constant number is in the exponent, like f(x)=2

Again you could fill in some typical x values, like -1, 0, 1, 2 etc.., and simply find out, that

the whole function is "shifted" to the left, by three points, compared to f(x)=2

If figure 1 above, you see that happening for f(x)=2

-You try that for f(x)=2

You can find more of such examples in note 7.

-Suppose you would have f(x) = -2

but this time mirrored with respect to the x-axis. Then it would be a decending function.

-Next, suppose we would have f(x) = b*2

Compared to 2

before the exponent, makes it rise more steeply, if "b" > 1.

All of the above, are just a few examples of socalled "transformations".

A slightly more general expression for an exponential fuction would be:

exponential power, also called the "base" or "base number" (like the "2" in 2

Again, even a tiny bit more general expression would be:

It has the same role as for example "2" in the equation f(x) = 2

The formula above, is absolutely no different from what we already have seen.

Usually, in math, we use "y" and "x" in our formula's.

However, if something changes with time "t", then we often simply use time as the variable,

(represented by the "t" symbol) to take the place of "x". It really does not matter.

In that case, we do not have an "x-axis", but instead we use a "t-axis".

Secondly, we might replace f(x) or "y" (which are the same), with "N", or "N(t)", if indeed

N is a number that is dependent on "t" (like the same roles like x and y had).

An example is the growth of a colony of bacteria, where the number "N" of bacteria,

depends exponentially on "t".

Especially in researches and articles from biology, sociology etc.., this notation is common.

Note that in a formula like N(t) = b g

the "start value" or "initial value". This is so, since when t=0 (or x=0), the exponential

resolves to "1". That is:

N(0) = b * g

Ofcourse, you remember that anything to power of "0", is simply "1". Thus g

Such initial value is most often interpreted as the start value when the observations begin,

like for example the growth of bacteria over hours, or days (or other time interval).

So, often the measurements start at "t=0". But, it could also be so, that folks start

in the year 2007 (starting year), and record the usage of Internet of the inhabitants

of a city. In such an example, a researcher could observe as of 2007, up to 2012.

Then, 2007 is ofcourse not really "t=0", but it is treated like so.

It's important to note that:

N = b, when t=0, which is the initial value (of N).

g is also called the base number, or "growthfactor".

Note 7, section 2, provides some examples of the practical use of the function N(t) = b g

Section 1 is just a small recap, of what you already could find in note 7.

The real objective of this note, is how to solve equations with exponential and/or logarithmic functions.

Thus, we need to explore logarithmic functions first.

From former notes, you may remember, that we may have the functions f(x) and g(x),

for which holds that f(g(x)) = g(f(x)) = x. If so, then both are called "the inverse" of the other.

That is, if we first apply f() on x, then apply g() on f(x), then we have x again!

In a graph in the XY coordinate system, then f() and g(), are each "mirrorred" in the line y = x.

Take a look at the figure below. It shows a few examples.

Figure 2. A few examples of inverse functions.

In the figure, you see on the right side, the function 2

function called "

The name "log()" may sound a bit weird right now, but I will deal with that in a moment.

Since both are each mirror function (thus mirrored in the line f(x)=x), they are called inverse functions.

So, then it should be true that 2

For the statement above, I should make it plausible right now, but I wait a small moment.

Generally, if "a" > 1, then a function like a

small values of "x". For example the function f(x)=10

x value of "6".

In some cases, if you mirror such a fast function in the line y=x, then you can still represent it

in a meaning full way, although ofcourse you are now talking about inverse values.

Nevertheless, sometimes the trick works.

Point 3 will indeed be thorougly demonstrated in section 5 below.

be each other inverse function?

That would be one of the cores, to justify it: The "log" is the inverse of the "exp".

Let's try:

Suppose f(x)=2

Then g(f(x))=

That is:

If:

g(f(x))=

Then:

2

And we only have applied the definition, to find this statement.

Not in all regions,exactly the same notation is used. Let's see for example

In some regions it might be written as:

That's not world shocking ofcourse, but it's important to know there are different notations,

which all means the same thing.

To get used in working with logarithms, here are a few examples with just numbers:

When you indeed have carefully studied those 3 examples, you should agree that:

The "base", or "base number" is the number that's lifted up, in an expression like

Maybe it's nice to show a part of the graph of f(x)=

the graph takes on positive values (when it gets above the x-axis), it still "rises", but slower and slower,

as x got larger.

Well, that is not strange at all. It's the inverse of 10

you can see that it is rising faster and faster, as x got larger.

Figure 3. Illustration of

In the graph, you can see that

As another illustration: what would be

To understand that, you might also consider it's inverse function: 10

In that case, we have for x=2, the value of 10

Hopefully can see it all work out, considering that both are mirrors with respect to the line y=x.

But you can also immediately refer to the definition:

If

The "10 base Logarithmic function" is often used in Calculus, also due to the Worldwide used

decimal number system.

at note 1, "arithmetic", where as of section 7, some basics are explained.

This section is extremely small. I just want to make sure, that working with

powers is quite comfortable for the reader.

We have seen exponential functions, like f(x)=2

These are real functions, and operate over ℝ, that is, the entire x-axis, or all values that

x can attain on the x-axis (the domain is ℝ, all real values).

In the following, we will talk just about numbers, represented by "a", or "b" etc..

We just want to know how to work with powers of different degrees.

So, for example, "a" can be any number like: 1, 5, 10, -1, -17, ⅓, 365, 0.15, 2.5 etc..

Here are some illustrative examples:

a

Example:

a

a^{p}- a ^{q} |
= | a^{p - q} |

Example:

a^{7}- a ^{5} |
= | a*a*a*a*a*a*a ---------------- a*a*a*a*a |
= | a*a | = | a^{2} |
= | a^{7-5} |

(a

Example:

(a

(ab)

Example:

(ab)

However, I like to make sure we are on the same line with "negative" powers,

so that's why we cover that in section 3.2 below:

then you would say, that's no problem, and the answer is: 5

When you see this: 5

then there is no problem too. Just follow rule 1: 5

Now, it's also important to realize, that an expression like:

1 -- 10 ^{3} |
= | 10^{-3} (note the negative exponent) |

A negative exponent, means that you may also write it as the denominator with the opposite sign of the exponent,

as we have seen above.

Now, if you see something like this:

1 -- 10 ^{-5} |

then it's really equivalent to 10

So, we can also simplify expressions like:

p^{4} * p^{-3}------- p ^{2} |
= | p^{4} * p^{-3} * p^{-2} = (just rule 1) => p^{4-3-2} = p^{-1} (note the negative exponent) |

Most often, the multiplication symbol (like * or x) is left out from expressions.

For example:

qp

Sure, such an expression as above, is quite nasty. However, it's fortunately no more than following the "rules".

(Who likes to follow the rules all the time?? Not me. However, with math..., it's another story. Follow the rules.)

This is actually a short section too. I just want to list some important properties

of logarithms.

The most important (defining) property is this one:

and, as another example:

The following list of properties, may be viewed as mathematical rules for working with logarithms.

However, we are going to review an example, to see if Rule 1 works:

Suppose we have a=2 and b=3, with a logarithmic base of 4:

(You might even take a look at the examples above.)

So:

Now let's see what the following will return:

Note: 4 * 4 * 4 * 4 * 4 = 16 * 4 * 4 * 4 = 64 * 4 * 4 = 256 * 4 = 1024.

Let's calculate the following. Suppose a=10000 and b=100:

Thus:

Now let's see what the following produces:

n *

However, in the next section, we will get a fair share of examples.

Ofcourse, there are more important properties. We will see those, when we need them

in coming examples.

I think we are ready for section 5: Solving equations with exponents and logarithms.

Let's just discuss a few important ones, and illustrate them, with one

or a few examples.

Most often, the equation has the variable "x" in the exponents, and "solving"

then means finding the x values for which the equation is true.

But that is not important. It all revolves around the method, thus on how we approach

the equations. That's the key !

Then we may immediately reduce the equation to:

Why? Well if we would see for example a

that would really fit that equation, is "x = 3". It's rather obvious ofcourse, given such

simple example. However, it's valid for complicated expressions as well.

Thus:

5

Let's checkit, by filling in "x=4" into the equation: 5

That's correct.

This is a nasty one. It may take some time, to find a usable approach. But, that's normal.

8^{(-2-4x)} |
= | 1 ------- 8 ^{(2x-4)} |

2

I deliberately skipped showing an intermediate step. It would be great if you could find that.

Now we have everything in the same base number (2). Thus:

-6-12x = -6x+12 => -6x=18 => x= -18/6.

6

The base numbers are already exactly the same. Thus:

x

-In many cases, the basic rule:

"

Or read it this way:

g

then:

the_exponent =

Or, ofcourse, we need to make use of the rules of section 4.

Thus:

5

-3 +

Thus:

5

5x = 25 => x = 5.

Solve the equation:

Set al the log's on one side:

3

x

x

Such a quadratic equation might have 0, 1, or 2 solutions.

You might see note 3, for solving such equation.

Solve:

We again apply a rule from section 4:

1 = (x-5)(x-1) = x

So, we need to solve the quadratic equation x

Fortunately, you know how to do that !

More examples will follow shortly!

f(x)=log(x)

Then you know that here, x must be larger than "0".

If you wonder why exactly, then:

- log functions are the inverse of the a

exponential functions, it holds that y=a

You know that the inverse functions are mirrored in the line y=x.

So, that's one reason.

-The most important (defining) property of logrithms this one:

For b

So, for f(x)=log(x), we can say that x must be larger than "0".

- What is the domain for this function?

- What are the intersection(s) with the x-axis, if any?

We must have: x

Let's first solve g(x)=x

For g(x)=x

So, the domain for (x)=log(x

The graph of f(x) looks like this:

Figure 4. Illustration of f(x)=log(x

Do not look at the points where f(x)=log(x

of f(x) with the x-axis. That is not the question, actually

So, the x interval [0,4] is not a valid "x" interval for f(x).

Here we must have log(x

log(x

x

Using the "abc formula":

x= 2+(√20)/2 or x=2-(√20)/2.

- What is the domain for this function?

- What are the intersection(s) with the x-axis, if any?

Same sort of story again:

We must have 8-2x > 0 => -2x > -8 => x < 4.

Ofcourse, you can fully "igone" the "6" constant in "6 + 3*log(8-2x)",

since that only represents an translation along the y-axis, of 6, with respect to "3*log(8-2x)".

g(x)=6 + 3*log(8-2x)=0 =>

3*log(8-2x) = -6 =>

log(8-2x) = -2 =>

8-2x = 10

-2x = -799/100 =>

x = 799/200 (is approximately about 4).