Ways to Solve Logarithmic and Exponential Equations

The new way we learned to solve logarithmic equations is exponentiation. Exponentiation acts as an inverse sine, or a square root to make it easier to solve an equation. Just like a square root takes away certain powers, so does exponentiation.

ex.

ex. x=2 b=10 a=100

Exponential Equations

One -to-One Proerties
a^x=a^y if and only if x=y
Logx=Logy if and only if x=y
Example
2^x=32
2^x=2^5
x=5

Inverse Properties
a^logx=x
loga^x=x
Example
e^x=7
lne^x=ln7
x=ln7

Strategies for Solving Exonential and Logarithmic Equations
1. Rewrite the given equation in a form to use the One-to-One properties of exponential or logarithmic functions.
2. Rewrite an exponential equation inlogarithmic form and applly the Inverse property of logarithmic functions.
3. Rewrite a logarithmic equation in exponential formand apply the Inverse Property of exponential functions.

Example
Solve 2(3^(2t-5))-4=11
2(3^(2t-5))-4=11 Write original equation
2(3^(2t-5))=15 Add 4 to each side
3^(2t-5)=15/2 Divide each side by 2
log'3 3^(2t-5)=log'3(15/2) Take log base 3 of each side
2t-5=log'3(15/2) Inverse Property
2t=5+log'3(7.5) Add 5 to each side
t=(5/2)+(1/2)log'3(7.5) Divide each side by 2
t~3.417 Use a calculator

Properties of Logs

Properties of Logarithms


The change of base formula: used so that you can change the base of a common log. By using it you are evaluating the logs to other bases. this basic formula can be used for base 10 and for natural log.


Here are two examples:
The first one is natural log
The second one is base 10

All of the log properties can be derived, but here they are:

*These properties can also be used for natural logs as well*



Now you can use the properties above to to either expand or condense functions. Here are some examples

Expanding:
Condensing:

3.1: Exponential Functions and Their Graphs

Exponential Functions:

f(x) = a^x

where a is greater than 0, not equal to one, and x is any real number





Exponential Graphs:

The graphs of all exponetial functions have similar characteristics:

• Domain: (-∞, ∞)
• Range: (0, ∞)
• y-intercept: (0, 1)
• No x-intercept
• Horizontal Asymptote of y=0
• Continuous

Note:

• A positive x indicates an increasing function
• A negative x indicates a decreasing function

Transformations of Exponential Graphs:
y = ab^(x-c) + d

• a -> causes the graph to vertically stretch or shrink; affects the y-intercept; if negative, causes a reflection in the x-axis
• c -> causes the graph to move left or right (left if +, right if -)
• d -> causes the graph to move up and down
• x -> a negative x causes the graph to reflect in the y-axis

The Natural Base e:

• e = 2.71828...
• e is called the natural base and its function is called the natural exponential function

f(x) = e^x

• Domain: (-∞, ∞)
• Range: (0, ∞)
• y-intercept: (0,1)

Rules of Exponents:

• a^x * a^y = a^(x+y)
• a^x / a^y = a^(x-y)
• a^-x = 1 / a^x

Compound Interest:

• Continuously Compounded Interest: A = Pe^rt
• Discretely Compounded Interest: A = P(1 + r/n)^nt
• A = present value
• P = principle value (how much money is invested)
• e = natural base
• r = annual rate of return *must convert into decimal form
• t = time in years
• n = number of compounding periods per year (ex. quarterly = 4, monthly = 12)

3.2 -- Logrithmic Functions

Rule to Everything

ex. log(1000) = 3 -----> 1000 = 10^3
Logrithmic Formy = log a (x)

D: (0,∞)
R: (-∞,∞)

Exponential Formx = a^y



Natural Log Functionlog e (x) = ln(x)

D:(-∞,∞)
R:(0,∞)
Ln Vs. Log Graph

1. Ln function is related to the log function, because the are reflected on the line y = x.

Law of Cosines

In 6.2 we learned about law of cosines.

From this triangle, we determined that:
sin A= y/c, therefore y= c * sin A
cos A= x/c, therefore x= c * cos A

From this information, we derived the law of cosines to be used with side-angle-side (SAS).

Finally, we learn the law of cosines for SAS:

Also, you can use this to find out b and c as well.

You can also use law of cosines with side-side-side (SSS).

We also learned about Heron's formula to figure out area. SSS is needed for this.

In Heron's formula, s is derived from the term "semi-perimeter" or half the perimeter. You determine s by:

Chapter 6

Oblique trianges are triangles that aren't right.  They can be obtuse or acute.

A.  Law of Sines

Deriving:

1. SinA = h/b         SinB = h/a
2. h = b sinA          h = a sinB
3. b sinA = a sin B
4. (sinA)/a = (sinB)/b

Formula:

Example:

a = 7     b = 8     A = 30 degrees

1. 7/(sin30) = 8/(sinX)
2. 7 sinX = 8 sin30
3. 7 sinX = 4
4. sinX = 4/7
5. arcsin(4/7) = X
6. X = 34.85 degrees

To use the law of sines you must have at least an angle and its opposite side.

B. New Area of a Triangle Formula

Deriving:

1. A = 1/2 b h
2. A = 1/2 c h
3. A = 1/2 c (b sin A)
4. A = 1/2 bc sin A

This a side-angle-side triangle.

Formula:

Example:

b = 7     c = 9     A = 20 degrees

1. A = 1/2 x 7 x 9 x sin20
2. A ~ 10.77 inches squared