Is It Possible to Prove Logarithms Using Equations?

[TheMathNoob]

Homework Statement

Proof x^log_by=y^log_bx

Homework Equations

log_b is a one to one function, that is, if log_bx=log_by
hint take the log in both sides of the equation and use the previous hint

The Attempt at a Solution

x^log_by=y^log_bx
log_by log x = log_bx logy
log_by / log_bx=logy/log x
log_by/x=log(y/x)

the book says that log(r) without any base indicates that the statement is true for any base so we can say that log(y/x)=log_by/x

so
log_by/x=log_by/x

if this is true then by the hint, x=y, so the original statement must be true

 

[andrewkirk]

the book says that log(r) without any base indicates that the statement is true for any base so we can say that log(y/x)=logby/x

What does that mean? 'log(r)' is not a statement - it contains no verb. When log is written without a base the usual convention is for it to mean log base e, although in older texts it is sometimes taken to mean log base 10.

The proof is short. Just apply the fact that $a^c\equiv e^{c\log a}$ to both sides and use the fact that $\log_d f=\frac{\log f}{\log d}$. Here I use the standard convention of using $\log$ for $\log_e$

 

[TheMathNoob]

What does that mean? 'log(r)' is not a statement - it contains no verb. When log is written without a base the usual convention is for it to mean log base e, although in older texts it is sometimes taken to mean log base 10.

The proof is short. Just apply the fact that $a^c\equiv e^{c\log a}$ to both sides and use the fact that $\log_d f=\frac{\log f}{\log d}$. Here I use the standard convention of using $\log$ for $\log_e$

look this is exactly what the book says , "When log(x) is used without any base being mentioned, it means that the statement is true for any base", I think that mine would make sense if apply log base b in both sides of the equation and then I go backwards.

 

[mfb]

log_by/x=log_by/x

That is a trivial statement because both sides are the same. You reduced the equation you have to show to something trivially true: done.
It is much easier to read the proof if you directly take the logarithm to base b.

if this is true then by the hint, x=y, so the original statement must be true

Do you really mean x=y? That doesn't have to be true.

 

[andrewkirk]

look this is exactly what the book says , "When log(x) is used without any base being mentioned, it means that the statement is true for any base",

That may work alright for some high-level statements, such as 'log z(u) is a linear function of u', but it's hard to think of any equations where it would work. It certainly doesn't work in this instance.

Ah, now I've worked out what the book means. The author is saying that she will only use the word 'log' without specifying a base when the statement she is making is true in all bases. That is guidance as to what she may do, not as to what you may do. When writing equations, the base needs to either be stated, or unambiguously implied (as per the e convention). The author's statement is a guide to interpreting her text, not a technique you may use in writing proofs.

 

[mfb]

A way to fix that: take the logarithm to base a, where a is an arbitrary positive real number.
On the other hand, and as I said before, if we want to choose a=b later, we can use log_b the whole time.

 

[TheMathNoob]

That is a trivial statement because both sides are the same. You reduced the equation you have to show to something trivially true: done.
It is much easier to read the proof if you directly take the logarithm to base b.

Do you really mean x=y? That doesn't have to be true.

so mine is right

A way to fix that: take the logarithm to base a, where a is an arbitrary positive real number.
On the other hand, and as I said before, if we want to choose a=b later, we can use log_b the whole time.

thank you so much, yes , I applied log base b at the beginning and came up with the trivial statement.

 

[TheMathNoob]

That may work alright for some high-level statements, such as 'log z(u) is a linear function of u', but it's hard to think of any equations where it would work. It certainly doesn't work in this instance.

Ah, now I've worked out what the book means. The author is saying that she will only use the word 'log' without specifying a base when the statement she is making is true in all bases. That is guidance as to what she may do, not as to what you may do. When writing equations, the base needs to either be stated, or unambiguously implied (as per the e convention). The author's statement is a guide to interpreting her text, not a technique you may use in writing proofs.

anyways, it is wrong. I made a stupid mistake of log properties xd.