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

#### Ted_Grendy

##### New member
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
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
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

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#### Denis

##### Senior Member
If log(u) = log(v) then u = v

#### Jomo

##### Elite Member
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
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
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.

I'm an American.

#### Jomo

##### Elite Member
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
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
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

$$\displaystyle log_a(x) = log_a(y) \implies x = y.$$

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