Another Log Problem: logY^3 = logx(x+4)^2

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Ted_Grendy

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Hello all

I have another log problem which I was hoping someone could help me with.

I have the following:-

logY^3 = logx(x+4)^2

I have been told that the above statement is the same as:-

Y^3 = x(x+4)

The bit I don't understand is where does the log go?
What law allows the log to be removed from both sides?

Can anyone help?

Thank you.

(assume base 10 log)
 
Ted_Grendy said:
Hello all

I have another log problem which I was hoping someone could help me with.

I have the following:-

logY^3 = logx(x+4)^2

I have been told that the above statement is the same as:-

Y^3 = x(x+4)^2

The bit I don't understand is where does the log go?
What law allows the log to be removed from both sides?

Can anyone help?

Thank you.

(assume base 10 log)
Click to expand...

If log A = log B then A = B.
 
Ted_Grendy said:
Hello all

I have another log problem which I was hoping someone could help me with.

I have the following:-

logY^3 = logx(x+4)^2

I have been told that the above statement is the same as:-

Y^3 = x(x+4)

The bit I don't understand is where does the log go?
What law allows the log to be removed from both sides?

Can anyone help?

Thank you.

(assume base 10 log)
Click to expand...
It sounds so good, I'll repeat it again: If log A = log B, then A = B
 
Last edited:
Ted_Grendy said:
Hello all

I have another log problem which I was hoping someone could help me with.

I have the following:-

logY^3 = logx(x+4)^2

I have been told that the above statement is the same as:-

Y^3 = x(x+4)

The bit I don't understand is where does the log go?
What law allows the log to be removed from both sides?

Can anyone help?

Thank you.

(assume base 10 log)
Click to expand...
Log (for any base) is an increasing function which means that if A < B, then it must be that Log(A) < Log(B). So if A and B are different values it can't be that Log(A) = Log(B). But if Log(A) = Log(B), then yes A = B. In fact if A=B and they are both positive, then Log(A) = Log(B)

Note: If you have something like Log(-2) = Log (x) we can NOT conclude that x=-2 because Log(-2) makes no sense (you can't take the log of 0 or a negative number).
 
Ted_Grendy said:
Hello all

I have another log problem which I was hoping someone could help me with.

I have the following:-

logY^3 = logx(x+4)^2

I have been told that the above statement is the same as:-

Y^3 = x(x+4)

The bit I don't understand is where does the log go?
What law allows the log to be removed from both sides?

Can anyone help?

Thank you.

(assume base 10 log)
Click to expand...
This question belongs in intermediate algebra or perhaps even pre-calculus, but it is an excellent question.

Logarithm functions, regardless of base, are members of a class of functions known as invertible. There is a theorem that proves

\(\displaystyle \text {If } f(x) \text { is invertible, then } f(x) = f(y) \implies x = y.\)

It can be shown that logs are invertible (in my youth, when logs were actually used for computations, we had to work with the inverse called an antilog).

But if you do not want to work through the proof that logs are invertible, a proof that is very advanced, you can just say that one of the laws of logarithms is

\(\displaystyle log_a(x) = log_a(y) \implies x = y.\)
 
Last edited:
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