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 :)

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