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

Ted_Grendy

New member
Joined
Nov 11, 2018
Messages
26
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)
 

Harry_the_cat

Full Member
Joined
Mar 16, 2016
Messages
939
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)
If log A = log B then A = B.
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
2,965
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)
It sounds so good, I'll repeat it again: If log A = log B, then A = B
 
Last edited:

Denis

Senior Member
Joined
Feb 17, 2004
Messages
1,436
If log(u) = log(v) then u = v
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
2,965
If log(u) = log(v) then u = v
You have start off slow with some students and use A's and B's before U's and V's. Didn't they teach you nuttin in business school
 

Denis

Senior Member
Joined
Feb 17, 2004
Messages
1,436
Didn't they teach you nuttin in business school
You forgot to end your sentence with a "?"
Didn't they teach you nuttin in grammar school
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
2,965
You forgot to end your sentence with a "?"
Didn't they teach you nuttin in grammar school
I'm an American. So no, they did not teach me anything in grammar school.
 

Denis

Senior Member
Joined
Feb 17, 2004
Messages
1,436

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
2,965
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)
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).
 

Harry_the_cat

Full Member
Joined
Mar 16, 2016
Messages
939
It sounds so good, I'll repeat it again: If log A = log B, then A = B
Jomo, If you "repeat it again" doesn't that mean that you are saying it for the nth time where n>=3 ?
 

JeffM

Elite Member
Joined
Sep 14, 2012
Messages
3,232
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)
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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