Natural Logarithms

Graph of a Natural Logarithms


Definition of Natural Logarithm

When

e ^y = x

Then base e logarithm of x is

ln(x) = log_e(x) = y

The e constant or Euler's number is:

e ≈ 2.71828183

Common Logarithms use the Base 10
Natural Logarithms use the Base e

Note:
All Natural Logarithms use the same laws and patterns as Common Logarithms
Division - Subtraction
Multiplication - Addition
Exponents - Move to the front
Same Base - Cancel
Change of Base - use e instead of 10

Natural logarithm rules

Product:

The rule name is product. The rule is,

logx (a.b) = logx a + logx b.

The example is,

logx (9.8) = logx 9 + logx 8.

Quotient:

The rule name is quotient. The rule is,

Log (a/b) = loga – logb.

The example is,

Log (10/8) = log10 – log8.

Change of base formula:

The rule name is change of base formula. The rule is

Log ab = logb /log a.

The example is,

Log 8.2 = log8 /log 2.

Power

The rule name is power. The rule is,

ln(a ^b) = b ∙ ln(a)

The example is,

ln(8⁵) = 5 ∙ ln(8)

Examples

Solve for x

	Divide both sides by 7
	Use Property of Logarithms, Part 2, to take the log of both sides
	Property of Logarithms:
	ln e = 1
	Divide both sides by 3
x » 1.266

	Use Property of Logarithms, Part 2, to take the log of both sides
(x + 2) ln 2 = (2x + 1) ln 3	Property of Logarithms:
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3	Distributive Property
x ln 2 - 2x ln 3 = ln 3 – 2 ln 2	Isolate terms with the variable on one side of the equation
x(ln 2 – 2 ln 3) = ln 3 – 2 ln 2	Factor out the common factor, x
x » 0.191

Exponential & Logarithmic Equations Part 2

Solving Logarithmic Equations steps:

1.   Place one logarithm with base 10 on each side of the equation.

2.   Using the Laws of Logarithms move the exponents the front of each logarithm. If the exponent contains a binomial we must use brackets around the exponent.

3.  If the exponent is in the brackets, multiply through brackets.

4.  Collect logarithms on one side and isolate the variable on the others. If there is more than one term containing the missing variable isolate these terms on one side and factor out the variable, then continue to isolate the variable.

5.    Solve for the variable using your calculator rounding to whatever decimal place the question instructs.

6.   Check your answer! Plug your answer back into the original question and check.

Example 1: Solve to the nearest ten-thousandth.

a, 2^x =45                          b, 5^x = 3 ^x-4

log 2^x =  log45                   log 5^x = log 3 ^x-4

xlog2 = log45                   xlog5 = (x-4) log3

x   = log₂45                    xlog5 = xlog3-4log3

x = 5.4918                      xlog5-xlog3 = -4log3

                                     x(log5-log3)= -4log3

                                    x    = (-4log3)/(log5-log3)

                                                x    = -8.6

c, 3^2x+1 = 100                               d, 3 ^x+1 = 7 ^x-1 .3
log 3^2x+1 = log100                  log 3 ^x+1 = log 7 ^x-1 .3
(2x+1)log3 = 2                    (x+1)log3 = (x-1) log7 +log3
2xlog3 +log3 =2              xlog3+log3 = xlog7 – log7 + log3
2xlog3 = 2 - log3            xlog3-xlog7 = log 3 – log3 – log 7
                                     x(log3 – log7) = -log7 X= 2 – log3                                                                                                                                                                                  x = -log7/ (log3- log7)
X = 1.5959                             x= 2.2966.

-JAN PHAM-

Natural Logarithms

     On Thursday we had a "test" on what we learned so far in Unit 4 and we also learned about Natural Logarithms. Natural Logarithms were created by Leonhard Euler in the 1700's, Euler was also the first one to use "f(x)" when working with functions. Natural Logarithms have a base e rather than a base 10 of Common Logarithms.

     Natural Logarithms have the same laws as Common Logarithms:
     Division - Subtraction
     Multiplication - Addition
     Exponents - Move to the front
     Same bases - Cancel
     Change of Base - use e instead of 10


     Here are some examples of what we learned:

Homework:
Expanding, Simplifying, Solving, and Evaluating Natural Logarithms - Questions 1 - 40

test

Mr p I am ready for the test, I have studied all night so I think ill need a sprite for energy in class thanks bye see you soon

unit 4 test

Hi mr. P I'm ready for the test hopefully its not too hard

Unit 4 Test

Hey Mr. P, I'm not over confident about the test today, but I guess I can pull through it and pass.

Unit test 4

Sorry that i have not been in class this week. For the test i think I'm Ready

test4.

Mr. P,
i think im ready for the test later. (:

Test#4

Mr.P
i think that I am ready for test tomorrow.

Just got home from work but yeah! Im ready for the test! :D

test

hi mr. p
im ready for the test tom. :)

Test

Yeah i think im ready too mr.p

Unit 4 Test

Hey Mr.P, I am ready for the test tomorrow.

Logarithmic Theorems II

Logarithmic Simplification Steps:

1. Rearrange terms-Move all negatives to the end

    2.   Exponent Law-Coeffcients become exponents

    3. Multiplication/Division Law-Addition becomes multiplication

       & subtraction becomes division

 4. Fractional exponents become roots (there are none in this example)

Change of Base Formula

Example:

Logarithmic Expansion

Today in class we learned how to expand logarithmic equations.

The steps for expanding Logarithmic equations:

1. Roots become fractional exponents.

2. Division Law - Division becomes subtraction.

3. Multiplication Law - Multiplication becomes addition.

4. Exponent Law - Exponents become coefficients (move to the front).

Logarithmic Functions Continued

So today we went over some examples in the unit 4 booklet. There are two types of graphing we learned about in class. We learned how to graph inverse logs and normal logs.

To find an inverse of f(x)=a^x using four steps method:
1-Replace f(x) with y
2-Reverse the roles of x and y
3-Solve for y in terms of x
4-Replace y with f(x)^-1

Here is an example below:

















Normal Log
- To graph a normal log all you do is make the table of values like you do for the others. You first graph the basic points and then you graph the shifted points. This is like unit 2 graphing.

Here is an example below:

Logarithmic Functions

Yesterday in class, we learned how to convert from a logarithmic form into exponential form.

log₁₀(100)=2

We then use the 7 rule

=100

EXPONENTIAL FUNCTIONS

On Wednesday Nov 10, we learnt about exponential functions.

The function f(x)= ab, exponent x, where a and b are real numbers such that "a" is not equal to zero, "b" is greater than zero and "b"is not equal to one,is an exponential function where :

"a" is a constant

"b" is a base

"x" is an exponent.

Basic curve : f(x)= a exponent x where a>o.
y= a exponent -x will reflect the basic curve in the y-axis.
y= -a exponent x will reflect the basic curve in the x-axis.
y= a (exponent x-h) + k will shift the basic curve.

An exponentail function is said to be an increasing function when a is greater then 1.

An exponential function is said to be an decreasing function when a is greater then 0 and less then 1.

Example No:1

Sketch the following exponential functions on the same cartesian plane.

a. f(x)= 5 exponent x b. g(x)= (1/5) exponent x c. h(x)= -5 exponent x




Example No: 2
Sketch the following exponential functions on the same Cartesian Plane.

a.f(x)= 4 exponent x b.p(x) = 4 exponent x+3 c. k(x)= 4 (exponent x) -2
d. q(x) 4 (exponent x-2) + 1

Unit test 3

Hey Mr. P. I'm feeling pretty good about the test today, the assignment and review really helped a lot to prepare for the test. See you in class.

Unit Test 3

Hiya Mr. P. To be honest I'm somewhat ready for the test but there are a few things i still don't understand like Sum and difference Identities. Lets hope for the best.

Unit 3 Test

Hey Mr. P, I'm somewhat ready for the test. I find some identities confusing.. but i should be fine, i guess. I'll just see when I get the test back. - katherine

unit test 3

Hi Mr. P,

actually, im not that ready for our test later. 

some identities are confusing.  i should be fine i guess since i did some review. ((:

Test 3

Hi Mr. P, I don't really know how I feel about the test I guess I will see how it goes once I get the test back

Unit 3 - test

Hey Mr. P im feeling pretty good about the test. Some of the identities are confusing but I should be ok.

Unit 3 Test

Hey Mr.P

Not going to lie , I'm not even close to ready!

TEST #3

Hello!!
MR. P I am ready for the test on tomorrow

Unit 3 Test

Hey Mr.P, I am ready for the test on friday, nothing is giving me trouble. If I go over the review I should be fine.

Double Angel Trigonometric Identities

Yesterday we learn in class Double Angel Trigonometric.

Sin(2Ө)=Sin(Ө+Ө)

=SinӨCosӨ + CosӨSinӨ

=2sinӨCosӨ

Cos(2Ө)=Cos(Ө+Ө)

=CosӨSinӨ - CosӨSinӨ

=Cos²Ө - Sin²Ө

Tan(2Ө)=Tan(Ө+Ө)

=TanӨ + TanӨ

1 - TanӨTanӨ

=2 TanӨ

1 - TanӨ

October Hall of Fame

And the winner is:
Bryden - 45 points - 5 bonus points
Tony - 45 points - 5 bonus points
Nikki T - 26 points - 4 bonus points
Michael - 21 points - 3 bonus points
Maria - 20 points - 2 bonus points
Cchristine - 20 points - 2 bonus points
Jordan - 12 points - 1 bonus point

Question about Assignment #3

Mr. P
What is the formula for

Sum and Difference Identities Part 2.

Last Friday, November 5, we learned more about sum and difference identities in different ways. This includes finding the coordinates of and the quadrant where it lies, verifying, finding exact values and expressing the following as function of .

EXAMPLE 1:


Step 1 : To figure out the points and we must identify the other point of the right triangle using the Pythagorean Thereom equation . Since is then...


In this case, the other point will equal to -15.



So the values will be...


Step 2: We must do the same step to figure out the values of and by using = . Since equals to then...

In this case, the other point will equal to -4.


So the values will be...



Now that we have all the values that we need to solve

Step 3 : We must solve for first. To do so, we must figure out the equation, which is...


Now all you have to do is plug in the values that equivalents the equation, like this...

Then multiply...

Which equals to...

So =

Step 4 : Now, we have to solve for . The equation for it is...


All we have to do now is plug in the values like the one we did from the previous equation, like this...

Then multiply...

Which equals to...

So =



The Quadrant in which it lies is in QII because cos = negative and sin = positive.