Expert: Sherry Wallin - 2/4/2010

Question
QUESTION: How would you solve this in terms of n, I have been trying to solve this for approximately 2 hours with no avail. It is for a friend, and her teacher said that it involves logarithms.

A = 800(1 + 0.1n)
A = 800(1+0.7)^n

ANSWER: Bob~
    Since you have labeled this problem 'intersection point' I assume you want to know the solution to the system where n is one of the variables and A is the other and since you have both equations written as A = ... that means that A is the dependent variable (vertical axis) and n is the independent variable (horizontal axis). I will solve each equation for n and let you do the rest:

first equation A = 800(1 + 0.1n) -> [A-800]/80 = n
What did I do? I multiplied the RHS getting 800 - 80n and then moved the 800 to the LHS and then divided the result by 80. (RHS = right hand side and LHS = left hand side)

second equation A = 800(1+0.7)^n: I will take the log of both sides of the equation getting
log A = log[800*(1.7)^n] since 1 + .07 is 1.07
the product of the logs is the sum of the logs so log[800*(1.7)^n] = log 800 + log(1.7)^n
the power on a log can come out in front of the log so log(1.7)^n becomes n*log(1.7)
you now have log A = log [800(1+0.7)^n] = log 800 + n*log(1.7) move log 800 to the LHS getting:
log A - log 800 = n*log(1.7) and divide both sides by log(1.7). (log (1.7) is just a number you can crunch on a calculator). The final result is that you have n = [log A - log 800]/log(1.7)

You now have two values for n, you can set them equal to each other and solve for A if you like...[A-800]/80 = [log A - log 800]/log(1.7)

Math Prof

---------- FOLLOW-UP ----------

QUESTION: Hi,

Thanks for responding. The solving that you did was the part that I had already progressed towards ( except I isolated for A instead of n). What I really wanted was the intersection point, because I have no idea how to solve an equation that involves the isolated variable polynomially and the isolated variable with a logarithm wrapped around it.

If I was to simplify further what you have isolated for with A, the equation would be A*log(1.7) - 800*log(1.7)= 80*log A - 80*log 800 . I don't understand how one would solve this equation for A, this was my original intention, not for the simplification of the intersection point. I'm sorry I phrased the question wrong.

You did however solve one of my other questions that I have been wondering for a while: when you log a quantity being multiplied by another such as 800(1.7)^n, which I am truly grateful for.

I hope to hear from you soon, I won't ask any further questions, thanks in advance for helping me out.

Bob

Answer
When I solved for n and got n = [log A - log 800]/log(1.7) I was solving for n (but A is still in it). I think I understand what it is that you want and it can't be done algebraically. When you have one variable to a power and that same variable as a factor you can't use algebra to separate them. You need to use numerical techniques, you must be studying numerical analysis. There are approximation techniques like Newton's law that you can use.

Math prof