## November 14, 2006

### Logarithm's Review

Hi guys, this is Jess and im supposed to have my scribe post to be done by yesterday but since my dad took away the computer and i had work after school till like 1030 so i couldn't have it up on time. Im sorry! Anyways, on monday, all we did in class with Mr. K were to review stuffs about logarithms, word problems and corrected our very first online quiz!
Here's the first one:

1. Which of the following best defines the inverse of ƒ(x) = e^x + 3?
f(x) = e^x + 3
y = e^x + 3
x = e^y + 3
x - 3 = e^y
ln (x-3) = y

2. If \$1000 is invested in an account that pays 6% per year, compounded quarterly, how much will be in the account at the end of 10 years?
A = ?
Ao = 1000
r = 0.06
n = 4 (quarterly)
t = 10

A = Ao (1 + (r/n))^nt
A = 1000 (1 + (0.06/4))^(4)(10)
A = 1000 (1.015)^40
A = 1000 (1.814018409)
A = 1814.02

3. How much should be invested in an account that pays 6% per year, compounded continuously, so that there will be \$10 000.00 in the account ten years from now?

Conpounded continuously formula: A = Pe^rt
A = 10 000
P = ?
r = 0.06
t = 10

10 000 = P(e)^(0.06)(10)
10 000 = P(e)^(0.6)
10 000 = P(1.8221188)
P = 5488.11

4. Ann’s initial investment doubles in 5 years. If interest is compounded continuously, at what rate did Anna invest her money?

A = 2
P = 1
r = ?
t = 5
and A = Pe^rt
2 = 1(e)^5r
2 = e^5r
ln (2) = 5r
(1/5) ln (2) = r

5. Suppose that a > 0. The graph of f(x) (illustrated) has a horizontal asymptote. What is the equation of the horizontal asymptote?

f(x) = b + e^-ax
y = b + e^-ax
b is the vertical shift
a is the horizonal compression
"-" reflects over the x-axis
=> horizontal asymptote y = b

6. What is the domain of f(x) = ln(2-x)?

x<2, and the argument have to be positive, not negative or either zero
therefore D(-∞,2)

7. The solution of 5^2x = 7 is (1/2) log base 5 to 7

5^2x = 7
2x log 5 = log 7
2x = log 7 / log 5
x = (1/2) (log7/log5)
* base on the " change of base law "
therefore (1/2) (log7/log5) = (1/2) log base 5 to 7 x

8. Solve for x: log(2x + 3) = 5

log(2x + 3) = 5
10^5 = 2x + 3
100 000 = 2x + 3
99997 = 2x
x = 99997/2

9.Which of the following expressions is equivalent to ln a / ln b

* Base on the " change of base law "
ln a / ln b = log base of b to a

10. Which of the following choices is equivalent to: e^ln x + ln y

e^ln x + ln y
e^ln(x)(y)
xy

11. If x > 0, then find an expression equivalent to:10^-2log(1/x)

10^-2log(1/x)
10^(log(1/x)^(-2))
10^(log x ^ 2)
x^2

And here's another word problem that we had on that class day.
An archearchaeologists finds a fossil that contains 60mg of Carbon-14. She knows the bone originally contained 100mg of Carbon-14. How old is the fossil if the half life of Carbon-14 is 5700 years?

A = 100
Ao = 60
m = o.5
t = ?
p = 5700

A = Ao(m)^t/p
100 = 60 (.5)^t/5700
0.6 = (1/2)^t/5700
ln (0.6) = t/5700 ln (1/2)
5700 ln (0.6) = t ln (1/2)
(5700 ln (0.6)) / ln (1/2) = t

And this was all we did on Monday. Once again, sorry for had it up late. Hope you guys did good on the test :). And by the way, thanks Steve for being the scribe for today's class!
And if i had done anything wrong in my scribe post, please let me know so i can fix it. Thanks!
Goodnight everyone :)