Thursday, April 17, 2008

The Natural Logarithm (Continued)

OUTLINE:

1st Session

• Properties of exponential functions
• Properties of logarithmic functions
• Change of base formula
2nd Session
• Simple interest versus compound interest
• Numeracy: how big is a million, billion, trillion, googol, and googolplex?
• e, ln, the exponential function, and the natural logarithm
• Solve equations using ln

PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS:
A SUMMARY

Refer to SLIDES 2 and 3, for exponential functions; 4 and 6, for logarithmic functions.

Definitions for the exponential and logarithmic functions.

Domain: All permissible x-values of a function on a graph.

Range: All permissible y-values of a function on a graph.

y-intercept: The y-coordinate when x = 0.

Increasing: Function with a graph that goes up as it's followed from left to right. (e.g. exponential growth)

Decreasing: Function with a graph that goes down as it's followed from left to right. (e.g. exponential decay)

Concavity
:
Concave up - Function is concave up if the graph is facing up the y-axis, like a cup having its opening facing up.
Concave down - Function is concave down if the graph is facing down the y-axis, like a bell having its opening facing down.

Asymptote: A line in which the function on a graph approaches infinitesimally closely but never meets.

CHANGE OF BASE FORMULA

This is a review of a previous class.

Refer to SLIDE 5 for a better visual of the proof of the change of base formula (how we derived the formula using an example and the definition of a logarithm), examples of the change of base formula, and the graph of the change of base formula in action.

log_2(x) = y

1. By definition, a logarithm can be expressed as this:
2^y = x

2. Find the log of both sides.
log(2^y) = log(x)

3. Use power law.
ylog2 = log(x)

4. Solve for y.
y = (log(x))/(log(2))
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This is an example using the change of base formula:
7^y = x
y = (log(x))/(log(7))

NUMERACY:
HOW BIG IS A MILLION, A BILLION, A TRILLION, GOOGOL, AND GOOGLEPLEX?

1 million seconds = 12 days
1 billion seconds = 32 years
1 trillion seconds = 32 000 years

Note: In some countries, such as Canada and the United States, a billion is a 1 followed by 9 zeros, and in Great Britain by 1 followed by 12 zeros. The above billion is the Canadian billion. Also, the above calculations are approximations.

We watched the One Billion is Big video:
http://numeracy.wikispaces.com/Online+Numeracy+Videos

googol = 10^100
googolplex = 10^googol = 10^(10^100)

We googled googol and realized that googol is a number that's greater than the number of the atoms in the observable universe.
http://en.wikipedia.org/wiki/Googol

e, ln, THE EXPONENTIAL FUNCTION, AND THE NATURAL LOGARITHM

Refer to SLIDE 8 for the calculation of e to 1 000 000 digits.

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Pronunciation of ln (the log_e) is found on SLIDE 9 as well.
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Refer to SLIDE 11 for the concept of ln(e) = 1.

ln(e) = 1

The ln_e, or "ln of e", is the "log of base e of e, " which is 1.

This is similar to log_2 (2) as being "the log of base 2 of 2 is 1."
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If we ever talk about THE exponential function or THE natural logarithm, then refer to SLIDE 9.

f(x) = e^x
...is the exponential function.

ln(x) = log_e(x)
...is the natural logarithm.

SIMPLE INTEREST VERSUS COMPOUND INTEREST

Simple interest is when you get interest from the original principal payment.
For example, Mr.K invests \$100 and has a 10% simple interest annually.
In Year A, he would receive 10% of the original payment of \$100, plus the \$100, which is \$110.
In Year B, he would receive 10% of the \$100 again, plus the original payment of \$100, which is another \$110.

Compound interest is when you get interest from the previous payment you received.

For example, Mr.K invests \$100 and has a 10% simple interest annually.
In Year A, he would receive 10% of \$100, plus the \$100, which is \$110.
In Year B, he would receive 10% of the \$110, plus the original principal payment of \$100

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Refer to SLIDE 7 for the calculations using the formula for the compound interest.

To calculate compound interest, use the formula,

A = P(1+r/n)^(tn)

whereas A is the amount, P is the principal (the money invested), r is the rate expressed as a decimal in the equation and not as a percent, n is the number of times taken, and t is the time (such as the number of years).

Note: Annually means once a year. Biannually means twice a year.
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Refer to SLIDE 10 for the calculations using the formula for the compound interest.

Another way of calculating the compound percent is to use the formula,

A = Pe^(rt)

whereas A is the amount, P is the principal, e is e--as in ln (the log of base e), r is the rate, and t is time.

SOLVING EQUATIONS USING LN

Refer to SLIDE 12 for solving a sample equation using ln.

Solve.

19^(x-5) = 3^(x+2)

1. Find the ln, which is the log of base e, of both sides.
ln19^(x-5) = ln3^(x+2)

2. Using the definition of a logarithm, we can rewrite the equation as...
(x-5)(ln19) = (x+2)(ln3)

3. Like solving for a simple jr. high math equation, we apply the same rules here using distribution.
xln19 - 5ln19 = xln3 + 2ln3

4. Like solving for a simple jr. high math equation, we check to see if we can collect like terms, but we can't, but we can put the variable x to the left side of the equation, thus making us closer to isolating x.
xln19 - xln3 = 5ln19 + 2ln3

5. Factor out the x.
x(ln19 - ln3) = 5ln19 + 2ln3

6. Solve for x.
x = (ln19 - ln3)/(5ln19 + 2ln3)

The second question on SLIDE 12 wasn't touched upon, but that can be done for homework, if you want, as preparation for tomorrow's lesson.

HOMEWORK:
Exercise 26: Natural Logarithms

Next scribe is nelsa.