Lesson # 14: Logarithm Equations & Identities

Today's lesson had two methods for solving equations, log= # and log=log

Method 1: Log = #                                                                                                                                             

Example:
log(2)x = 3                  1) Put into (b)^E = A form

2^3 = x

x = 8 

Example:

log(5)(x-3)+ log(5)x = log(5)10

log(5)x(x-3) = log(5)10               1) Put into B^e = A form

         x(x-3) = 10

       x^2- 3x = 10          2) Multiply both sides by x

  x^2-3x-10 = 0

   (x-5)(x+2) = 0        3) Simplify

           x = 5, -2     *-2 rejects*

           x = 5

*Note: negative answers are tricky, they must become positive in original equation or are rejected.*

       

Method 2: log = log                                                                                                                       

log5(x-3) + log5x = log5 10     1) Create only one log

log5(x-3)x = log5 10          2) Drop logs

(x-3)x = 10

x^2 - 3x = 10

x^2 -3x -10 = 0

(x-5)(x+2)

x= 5, -2            *-2 rejects*

x=5

Identities:

Equation        vs.         Identity

2x+3 = 5                    x + x = 2x

x=1                            x = all real numbers

x= specific #                rejects allowed

Prove the identity and state the value(s) of x for which it is true: (make both sides equal)

  

logx + log(x +3) = log(x^2 +3x)

  LEFT SIDE        RIGHT SIDE

Left side:  logx(x+3)

               logx^2 + 3x = Right side

*x must be great the 0     x>0*

Lesson # 13: Exponential Equations

In this lesson there are 2 different methods to find x: Common Bases method, or the Log method

Method 1- Common Bases:

3^2x = 27^3x-1              1) Find Common Base
3^2x = (3^3)^ 3x-1
3^2x = 3^9x-3                2) Drop the Base
2x = 9x-3
3 = 7x                           3) Solve
3/7 = x

Check:
3^3/7 = 27^3(3/7) -1
2.55 = 2.55  
correct!

Method 2- Logs:

3^2x = 27^3x-1                                  1) Log both sides
log3^2x = log27^3x-1
2xlog3 = (3x-1)log27                          2) Factor out brackets
2xlog3= 3xlog27 - log27
2xlog3- 3xlog27= -log27                     3) Factor Out x
x ( 2log3 - 3log27)  =      -log27          4) Solve for x
    (2log3- 3log27)     (2log3- 3log27)
x= 0.42857
x= 3/7

*Note: when using calculator, remember to use brackets or the answer may be wrong*

Lesson # 12: Laws of Logarithms

The Laws of Logarithms:

1) log(a)(xy) = log(a)x + log(a)y
2) log(a)(x/y) = log(a)x - log(a)y
3) log(a)(x^n) = nlog(a)x
4) log(a)n*root*(x)^1 = log(a)x^1/n or 1/nlog(a)x
5) log(a)b = log(b)/log(a)

Example: Express log8 in otherways
- log2^3 = 3log(2)
- log(2x2x2) = log2 + log(2) + log(2)

Example: Express 5 as a power of 2     *Log Both Sides!*
2^x = 5
log2^x = log5
xlog2 = log5
  log2    log2
x = 2.32

Lesson # 11: Exponentials & Logarithms

Exponentials vs. Logarithms:

Exponentials:      3^x                                      log(b) A=E <==> (b)^E=A
Logarithms:        log^x                            
                                                                 b=base     A=answer   E=exponent 

Example of Exponential to Logarithm Form:
1) 4^3 = 64   <==>  log(4)64 = 3
2) 2^-4 = 1/16  <==>  log(2)1/16 = -4

A logarithm is.... the exponent with a base of 10.

*To get the answer of a logarithm with a base of 10 is to count the zeros!*
Example:
log(10)1,000 = 3

Converting to base of x:

Example:     *base of 2*                     
1) 8^3x+1
(2^3)^3x+1
2^9x+3

2) 1/32^x-1
(2^-5)^x-1
2^-5x+5

Lesson # 7+8: Practice Tests

On the first day we wrote an in class practice test for chapter one.  We reviewed the test the next day, and got our marks back.  We were also given another test to take home to practice with over the weekend.

Lesson # 6: Absolute Value Tool

How to use the Absolute Value Tool:                    *y = f(x) to y = |f(x)|*

For the positive y values do not do anything. But for the negative values, they reflect over the x-axis.

Example:
base graph --- > y = 2x+3                   *All negative values become positive values*
absolute value graph ---> y = |2x+3|

How to use the Reciprocal Tool:                           *y = f(x) to y = 1/f(x)*

Steps:
1) Base
2) A.V. (zeros)
3) invarient points (ones)
4) reciprocal height

Lesson # 5: Combinations

Combinations:                       *Reflections, Shifts and Dilations*

If you have a dilation or reflection with a translation => FACTOR IT!

Example:                   *Factor only if needed*
y = *root*(2x) which makes it... horizontally compress 1/2x
y = *root*(2x+6) but you would factor it out, which makes it... y = *root*(2(x+6))

*Do stretch and reflections first then do the slides*

List the transformations:

y = 1/2f(-3(x+1))-4          
   
      Vert.                      Hor.
comp by 1/2x      reflect over y-axis
    down 4              comp by 1/3x
                                    left 1

Lesson # 4: Dilations

Stretching or Compressing:

y = f(x) goes to ky = f(x)        *This will make the equation, vertically stretch or compress by 1/k*

y = f(x) goes to y = f(kx)        *This will make the equation, horizontally stretch or compress by 1/k*
The way I remember these is when... if it's a whole number it will be compressed.
                                                      if it's a fraction it will be stretched.

Example:
1) y = f(1/2x) goes to x = 1/2x   which makes a... horizontal stretch x2
2) 2/3y = f(x) goes to y = 2/3x   which makes a... vertical stretch x3/2 or x1.5

Lesson # 3: Relections

Rules for x and y-axis:

For y = f(x) to go to y = f(-x), the graph of f(x) is reflected over the y-axis. In other words, x is replaced by -x.
For y = f(x) to go to -y = f(x), the graph of y = f(x) is reflected over the x-axis. In other words, y is replaced by -y.

Examples of basic equations:

1) y = (-x)^2 is the same as y = x^2, but it reflects over the y-axis.
2) y = -(x^2) is the same as y = x^2, but it refelcts over the x-axis.
3) y = 1/x is the same as y = 1/-x, but it reflects over the y-axis.
4) -y = *root*(16-x^2) is the same as y = *root*(16-x^2), but it reflects over the x-axis.

Rules for Inverses:     *An inverse is the reflection across the line y = x.*

1) Swap x and y                 y = f(x) goes to y = f^-1(x)
2) Isolate y
3) rewrite f^-1(x)

Example:               *Find f^-1(x)*
f(x) = -1/2x+3
x = -1/2x+3
(Multiply everything by 2)
2x = -y+6
y = -2x+6
f^-1(x) = -2x+6     *Remember that y = f(x) and f^-1(x) and NOT 1/f(x)*

Lesson # 2: Translations

Horizontal Translations   

For y = f(x) to go to y = f(x-k), the graph of f(x) shifts to the right by k if k>0, and it shifts to the left by k if k<0.
*A trick I use to remember this is, if the equation is (x-k) it will move to the right by k because it will become possitive.  And if the the equation is (x+k) it will move to the left by k. It will become negative.  So since the the graph is horizontal, the points will only move left to right.*

Example:        y = |x|                                                                        y = |x-4|

Vertical Transaltions

For y = f(x) to go to y-k = f(x), the graph of f(x) shifts up by k if k>0, and it shifts down by k if k<0.
*A trick I use to remember this is, if the equation is y-3 = f(x) it with move up by 3 because it will become possitive.  And if the equation is y+4 = f(x) it will move down because it will become possitive. It would be the same for y = f(x)+3 it would move up 3, and for y = f(x)-5 it would would move down by 5.*

Example:         y = x^2 +2

Lesson # 1: Functions You Should Know

1) Linear   y = mx+ b   (m = slope, b = y-int)                             

Example: y = 1/2x+2
    rise = +1
    run = +2                                                             
    y-int = +2

                                                                                      

2) Quadratic   y = (x+2)^2 -1                                                    Parabola = *root* (-2,-1)
                             

3) Cubics   *Deg 3* 

          

 y = (x+2)(x-3)(x-4)                                  Leading Term = x^3....+24

 zeros =-2, +3, +4

 y = -(x+1)^2(x-2)                                    Leading Term = -x^3....+2
 zeros =-1, -1, +2

*The negative makes it flip vertically*

4) Quartics   *Deg 4*                                        

                         
y = (x+1)^2 (x-2)(x-3)                                    Leading Term = x^4....+6

                        

5) Reciprocal   y = x  >> y = 1/x                      

a) draw base

b) vertical asymptote (zeros)

c) imvarient

d) reciprocal heights

                                                        

                                                       

6) Circles   r = 2                                              

 x^2+y^2 = 4

             y = 4- x^2

             y = *root*(4- x^2)       *top half*

             y = -*root*(4- x^2)     *bottem half*

                                                      

                                                         

7) Absolute Value   y = |x|                                                                         



|5| = 5

|0| = 0

|-2| = 2                                    

                                                    

8) Sqaure Root   y = *root* (x)        

             

 X   Y

 0    0

 1    1

 4    2

 9    3

-4   n/a            

                                              

9) Exponential   y = 2x

           

 X   Y

 0    1

 1    2

 2    4

 4    8 

-1  1/2

-2  1/4