6.4 Power of Law of Logarithms

Hey

Today we learned about Logarithms and the Power Laws.

The basic properties of logarithms are
e.g. log(b) 1 = - <-------> b^0 = 1

The Laws of Logarithms are as follows.
note* log base "a"(or any variable) = log(a)
1. Log(a) mn = log(a)m + log(a)n

Proof:
let: m=a^x, n=a^y
log(a)m=x, log(a)n=y

so:
mn = a^x multiplied by a^y
mn= a^(x+y)
which means that:
log(a) mn = x+y
therefore,
log(a) mn = log(a)m + log(a)n

2. log(a) (m/n) = log(a)m - log(a)n, note n cannot equal zero

let: m=a^x, n=a^y
log(a)m=x, log(a)n=y

so, m/n = a^x/a^y
m/n = a^(x-y)
log(a) (m/n) = x-y
log (a) (m/n) = log(a)m - log(a)n

3. log(a) (n^m) = mlog(a)n

let n=a^x

log(a)n = x
n^m(a^x)^m
n^m = a^(xm)
log(a) (n^m)= xm
log(a) (n^m) = mlog(a)n

4. Change of Base Formula
log(b)X = log(a)X/log(a)b
e.g. a)
3^x = 5
log(3)5 = x
x = log5/log3
x =(approx) 1.4650

Therefore, these following formula's outlines the logarithm laws.

6.3 Transfomations on Lagarithmic Functions

In class on Friday we learned how to graph and apply transformations to a logarithmic function. These transformations are very similar to the ones that we learned earlier in the year when we were dealing with the sin and cos graphs. The set up of these functions is the same as the set up for the log functions, y=alog[k(x-d)]+c, the variables of these functions affect the same parts of the graph as well. The a value will vertically stretch and compress it, the k value will horizontally stretch or compress it, the h will translate the graph horizontally and lastly the c value will shift it vertically.

*REMEMBER: the d value will shift the vertical asymptote and the c value will shift the horizontal asymptote, as well as the graph.

When applying transformations to a function follow these 5 steps…
Step 1: Ensure that the function is in y=alog[k(x-y)]+c form
Step 2: Apply vertical stretches, compressions and reflections
Step 3: Apply horizontal stretches, compressions and reflections
Step 4: Apply horizontal translations
Step 5: Apply vertical translations

From these 5 steps you should be able to create a mapping rule in which you can use to graph the function. (x,y) = (1/k x-d,ay+c)

You can now use the mapping rule to create a table of values so that you can graph the translated function.

*REMEMBER: a logarithmic graph is the opposite of an exponential graph so you must exchange the x and y values in the table of values.

Example:
x y = 10^x
-2 1/100
-1 1/10
0 1
1 10
2 100

x y= log10x
1/100 -2
1/10 -1
1 0
10 1
100 2
(then take the values from this table and apply them in your mapping rule to find the points so that you can graph your function)

The last things that we covered on Friday were the key features of a graph. Well, domain, range, and the asymptotes, can be found easily by looking at the graph, but how do we find an intercept when the point is not on the table of values? Well, just like in previous chapter what we do is solve for the variable using the log function. For example, if you were looking for the x-intercept you would make y=0 and then you could solve for x, we know this because at the intercept y=0. The opposite can be done for the y-intercept, x=0.

Period 1: 6.3- Transformations

Hi everyone :)

so on Friday we learned about transformations to logarithmic graphs, pretty much taking all our knowledge of transformations and applying it to a new type of graph. We have to remember a few things in order to graph correctly.

STEPS to apply multiple transformations

1. Ensure the function is in the form of f(x) = a log [k(x-d)] + c
***don't fall for the trap where the equation is not in factored form!! it will be on our test and exam so watch out for it!!***
2. apply vertical stretches, compressions, and reflections
3. apply horizontal stretches, compressions, and reflections
4. apply horizontal translations
5. apply vertical translations

mapping rule always begins with (x,y)--->

Note: We must keep in mind that the vertical asymptote depends on the horizontal shift of the equation. Any other stretch or compression, or vertical shift on the graph will not affect the vertical asymptote.

A logarithmic graph's key features:
- the domain is affected because of the asymptotes
- there is no horizontal asymptote
- x-intercepts can be found my solving for x when y=0 (using skills learnt in 6.2 with logarithms) for example:

y=-2log[1/2(x+6)]-3
0=-2log[1/2(x+6)]-3
-3/2=log[1/2(x+6)]
10^-3/2= [1/2(x+6)]
0.03162/.5 = x+6
0.06325 = x+6
-5.9368 = x

therefore, -5.9368 is the x-intercept.

- the table of values are different for this type of graph:
we must make 3 different tables in order to graph
the first one is our base exponential function
the second one is our base parent function of the logarithmic function
the third one is our table of values with all the transformations applied

6.2: Logarithms

A logarithm of a number is the value of the exponent to which the base must be raised to produce a given number. For example: log3(81)=4 because 3^4=81.





A logarithm is the inverse of a function, where the x and y values are switched. Therefore if y=b^x, the inverse of this function is x=b^y which is represented by y= logb(x). y equals the logarithm of x to the base b. The logarithmic function is useful for solving unknown exponents.

The graph displays the exponential function y= 2^x, along with its inverse, y= log2(x). They reflect on the y=x axis.

To write an exponential equation in logarithmic form we...

16 = 2^4

4 = log2 (16)

We read this by saying "4 equals the logarithm of 16 to the base 2".

To write a logarithmic equation in exponential form we ...

log3 (81)

let y = log3 (81)

Then, 3^y = 81

3^y = 3^4

y= 4

NOTE: common logarithms are logarithms with a base of 10. Many times questions will not state the base of 10 because it is understood that the base is 10.

This video will help you further understand logarithms... it sure helped me!

6.2 Logarithms

The concept of logarithms was created by John Napier, in order to simplify math, specifically for the use of astronomers. Previously, the only way to solve exponential equations, such as 25=10^x, was using the "guess & check" method. Now, logarithms can be used to solve such equations much more easily and provide an exact answer.

A logarithm of a function is the inverse of the function. Essentially, the input and output of the equation have switched roles.

For the equation: y=b^x, the inverse is: x=b^y, which is represented as y=logbx. This means that y is equal to the log base b of x.

The above graph displays the relationship between a function and the logarithm of that function. The two functions are reflections of each other about the y=x graph.

The following are examples of the relationship between the exponential and logarithmic forms of equations:

Logarithms to the base of 10 are known as common logarithms. Most of the logarithms we will encounter in this chapter are common ones, and these are the only logarithms that we can currently solve using calculators. When writing a common logarithm, it is not necessary to write the base number (10).

For example:

log27 = log₁₀27

The following are examples of common logarithms:

***Hint: need to understand/apply this for the thinking/inquiry section on quiz***

LOGARITHMS_ 6.2

The inverse of a function is what we now know as logarithm. Logarithm, which many people fail to understand fully, is simply the power or exponent (y) to which the given base must be raised to in order to produce a known value.
For example,
When solving for 10 ^y= 100, we know that in order to get 100, 10 must be multiplied to itself two times; 10*10 = 100, in other words, 10^ 2 = 100.
But when solving for an equation such as 10^x=32, trial and error could be quite exhausting to work with and so we use John Napier’s invention of the LOGARITHM, which simply states that if x=b^y, then y=logb(x).
Using logarithm, we can now state the exponential form, 10^2=100, in the logarithmic form, log(10)100=2 and the exponential form, 10^x=32, in the logarithmic form, log10(32)=x.
Often, however, when speaking of a base 10 in logarithm, the base value will be written as log, and will not be stated as it is understood to mean the same as log10. Such logarithms with a base 10 are known as common logarithms.

Below is a graph of a y=10^x function show in red and its inverse shown in blue. As you can see there is a strong relationship with y=1.5 as shown with the black line, and x=10^y, also known as y=log(x), as they intersect at a point which represent the x-value.

When we think of an inverse function we think of the x and y value switching places, similarly, when using log, an easier way to understanding the concept at the very basic level before growing further into this unit is to see it as the same; a switch in the positions of the x and y value. Such as the inverse of y=b^x which is x=b^y where you can see the x and y have switched positions, using the logarithm, we can similarly see that when we simply switch the positions of the inverse function we have, y=logbx, where b represents the base value.

EXAMPLE 1:
The exponential form, 4^3=64 can be written in the logarithmic form as
Log464=3 (recognise how the values highlighted in blue representing x and green representing y switch places)
EXAMPLE 2:
The logarithmic form, log2(64) can be written in the exponential form as
2*x=64
àUsing previous experience and knowledge, we know that 2^6 = 64, so evaluating the logarithm we get...
X=6

As we have seen, sometimes the exponents are not very easy to solve for, and so we turn to our calculators. So far, we have only learned to solve for base 10.
When using our calculators to solve for base 10 functions that we cannot solve on our own, we simply press LOG on our calculators, and type in the equation as we see it; so for instance if we were solving for log 0.001 (remember, this is like saying log10(0.001)), we simply PRESS log, and then type in 0.001; the calculator assumes that we are working with base 10 in such a scenario.
- Michelle Joseph

6.1 The Exponential Function and its Inverse

Hey guys,

So we are starting a new unit! This year is not going to be much different then what you have been learning in Grade 11.  Now, you will learn this later on but as an introduction - the main difference between what you will learn this year and last year is this:

Grade 11 Exponential Functions:

Solving for x when you have two exponential functions with the same base

ie.

27=3^x
3³=3^x (general form of: a^x=a^x)

Since we now have the same base, we can now only deal with the exponents.

3=x

or

y=b^x we are able to solve for x

Grade 12 Exponential Functions:

What if we have to solve for a^x = b^x ? This year, we will be learning how to solve for eponential equations that have different bases.

or in this unit since we will be dealing with inverse functions;

x=b^y is the inverse of y=b^x but how are we going to solve for y in the inverse exponential equation?

Just something for you guys to think about : )