How do you solve this? -33.1(1.04)^x - (0.75)^{x-1} + 37.5 = 0

[davidmazzei]

Im studying for my engineering license and I dont remember how to do math (oh the irony). This is an engineering economics problem that ends up having x's in the exponent's and I keep getting the wrong answer. See below for my work.

Thank you!!

 

[stapel]

Im studying for my engineering license and I dont remember how to do math (oh the irony). This is an engineering economics problem that ends up having x's in the exponent's and I keep getting the wrong answer. See below for my work.

Thank you!!

View attachment 36019

You can check your answer to any "solving" problem by plugging your answer back into the original exercise. If you plug [imath]2.77[/imath] in for [imath]x[/imath], what do you get?

It looks like you might benefit from reviewing the basics of exponents; you can find many lessons here.

Once you've refreshed, note the rules that say [imath]m^{a+b} = m^a\cdot m^b[/imath] and [imath]m^{-a} = \frac{1}{m^a}[/imath].

 

[Dr.Peterson]

Im studying for my engineering license and I dont remember how to do math (oh the irony). This is an engineering economics problem that ends up having x's in the exponent's and I keep getting the wrong answer. See below for my work.

Thank you!!

View attachment 36019

As I understand it, you got 3.58, but your source says 2.77?

First, it may be encouraging to know that what you have "forgotten" is something you will probably not have learned: This type of equation can't be solved algebraically! So, no, you shouldn't know how to solve it.

Except, of course, clearly someone expects you to. It can be done (numerically rather than algebraically) with technology (such as a graphing calculator) or with various hand methods of approximation. Have you learned anything like that? Are such tools on the list of what you are being tested on?

But you also need to review properties of logarithms. This line is wrong for several reasons:

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When you take the log of both sides, you have to follow rules to simplify; and there is no rule that allows you to take the log of a sum. You can't just take the log of each term, as you have done. You also didn't handle 33.1 correctly here.

But, again, correcting those errors won't make this method work.

I got the answer of (about) 2.77 (and another as well; perhaps you were told x has to be positive?) from Wolfram Alpha.

 

[Steven G]

...and there is no rule that allows you to take the log of a sum

Really? I didn't know that you even needed a rule to take the log of a sum. Personally, I just compute the log of the sum.

 

[khansaheb]

Im studying for my engineering license and I dont remember how to do math (oh the irony). This is an engineering economics problem that ends up having x's in the exponent's and I keep getting the wrong answer. See below for my work.

Thank you!!

View attachment 36019

Can you differentiate the LHS of the given equation?

 

[khansaheb]

Really? I didn't know that you even needed a rule to take the log of a sum.

I am assuming you are saying that with your tongue firmly planted in your cheek.

Just for clarification, I think the "law" being referred to is the "law of log of a product" where

Log_a(b*c) = Log_ab + Log_ac .....(with usual restrictions)

There is no "corresponding" law for Log_a(b+c)

 

[Steven G]

I am assuming you are saying that with your tongue firmly planted in your cheek.
Just for clarification, I think the "law" being referred to is the "law of log of a product" where
Log_a(b*c) = Log_ab + Log_ac .....(with usual restrictions)
There is no "corresponding" law for Log_a(b+c)

I question that. If d=b+c, then log_a(b+c) = log_ad. That is the rule, just add b and c and compute the log of that.
Dr Peterson clearly that that there is no rule that allows you to take the log of a sum. To me that means that you can't compute the log of a sum. After all, if you could, then there would be a rule.

 

[Dr.Peterson]

Really? I didn't know that you even needed a rule to take the log of a sum. Personally, I just compute the log of the sum.

I took what you said to be a joke (tongue in cheek, that is), pretending not to recognize slightly informal language.

It should be clear that when I said,

When you take the log of both sides, you have to follow rules to simplify; and there is no rule that allows you to take the log of a sum. You can't just take the log of each term, as you have done.

I meant

When you simplify the log of both sides, you have to follow rules to simplify; and there is no rule that allows you to simplify the log of a sum. You can't just take the log of each term, as you have done.

"Take", in context, doesn't mean "calculate" -- that isn't what he was doing! He was rewriting an expression. And there is no rule that allows you to rewrite the log of a sum as a sum of logs.

I didn't respond to your comment initially, because I took it lightly, and I don't think overreaction is helpful. But now you seem to be taking what you said too seriously, so it needs a response.

I question that. If d=b+c, then log_a(b+c) = log_ad. That is the rule, just add b and c and compute the log of that.
Dr Peterson clearly [said] that there is no rule that allows you to take the log of a sum. To me that means that you can't compute the log of a sum. After all, if you could, then there would be a rule.

Are you saying that computing the log is a "rule" corresponding to the rule he quoted, "Log_a(b*c) = Log_ab + Log_ac" ? I don't think so.

Please lighten up.