Expert: Paul Klarreich - 7/16/2007

Question
I'm having some difficulties figuring out these questions. Could you explain them to me? Thank you!

1.) Evaluate (1+1/x)^x   for x = 100, 10,000 and 1,000,000.    Describe what happens to the expression as x increases - be specific.

2.)Let f(x) = 62 + 35log(x – 4) represent the percentage of adult height attained by a girl who is x years old (from 5 to 15).Suppose that a girl is 4 feet 6 inches at age 10.    Determine how tall she can expect to be as an adult.

3.)If log3 = A and log7 = B, find log 7 9 in terms of A and B.

4.) Write e in8x^5-in2x^2  as a single term that does not contain a logarithm.

5.) Exponential functions and logarithmic functions are inverse functions.    It is because of this that it would make sense that they have some similar properties.    Identify which property of exponents is similar to  logAB  =  logA  +  logB  and which is similar to  logA /B =  logA  –  logB .Explain how they are similar.

I was able to work through the rest, but these ones I just couldn't figure out. Thanks for your help in advance!  

Answer
Questioner:   Jillian Whitney
Category:  Advanced Math
 
Subject:  Re: Exponential and Logarithmic Functions
Question:  I'm having some difficulties figuring out these questions. Could you explain them to me? Thank you!

1.) Evaluate (1+1/x)^x   for x = 100, 10,000 and 1,000,000.    Describe what happens to the expression as x increases - be specific.

2.)Let f(x) = 62 + 35log(x - 4) represent the percentage of adult height attained by a girl who is x years old (from 5 to 15).Suppose that a girl is 4 feet 6 inches at age 10.    Determine how tall she can expect to be as an adult.

3.)If log3 = A and log7 = B, find log 7 * 9 in terms of A and B.

4.) Write e^(ln8x^5-ln2x^2)  as a single term that does not contain a logarithm.

5.) Exponential functions and logarithmic functions are inverse functions.    It is because of this that it would make sense that they have some similar properties.    Identify which property of exponents is similar to  logAB  =  logA  +  logB  and which is similar to  logA /B =  logA - logB .  Explain how they are similar.

I was able to work through the rest, but these ones I just couldn't figure out. Thanks for your help in advance!

................................................
Hi, Jillian,

1.) Evaluate (1+1/x)^x   for x = 100, 10,000 and 1,000,000.    Describe what happens to the expression as x increases - be specific.

This is a well-known limit.  Your calculations should give values around  2.7, and:

Lim    (1 + 1/x)^x =  e, the base of natural logarithms.
x->inf

To prove this, you need calculus, which I don't think you are studying yet.

................................
2.)Let f(x) = 62 + 35 log(x - 4) represent the percentage of adult height attained by a girl who is x years old (from 5 to 15).  Suppose that a girl is 4 feet 6 inches at age 10.    Determine how tall she can expect to be as an adult.

Let  H(x) be the actual height at any age x, where x is in [5,15]
Let  AH be the height as an adult.

H(x) = AH * f(x) /100

We are given the facts that

At age 10,  H(10) = 4.5 feet.

4.5 = AH * f(10) / 100

4.5 = AH * [62 + 35 log(10 - 4)] / 100

You can do the rest, using your calculator.

AH = 450 / [62 + 35 log(10 - 4)]
................................

3.)If log3 = A and log7 = B, find log 7 * 9 in terms of A and B.

You wrote 7 9, which I don't think meant 79.  

log 7 * 9 = log 7 + log 9 =
log 7 + log 3^2 =
log 7 + 2 log 3 =
B + 2A

.................................
4.) Write e^(ln8x^5-ln2x^2)  as a single term that does not contain a logarithm.

Use the property that  e^ln(A) = A.

ln (8x^5) - ln (2x^2) =

   8x^5
ln ------ =
   2x^2

ln 4x^3

Now  e^(ln 4x^3) = 4x^3
..............................

5.) Exponential functions and logarithmic functions are inverse functions.    It is because of this that it would make sense that they have some similar properties.    Identify which property of exponents is similar to  logAB  =  logA  +  logB  and which is similar to  logA /B =  logA - logB .  Explain how they are similar.

logAB  =  logA  +  logB

Let  X = log A  and  Y = log B,  using base S

Then  A = S^X   and  B = S^Y, and further:

AB = S^X S^Y =  S^(X + Y)

That's it -- the rule for multiplying powers of the same base:

I think you can do the other one yourself.