Proof of logarithmic properties.

[pollytree]

Homework Statement

There are two log properties that I have to prove:
1) Explain why ln(b^1/n)=(1/n)ln(b) for b>0, set b=aⁿ

2) Explain why ln(a^r)=rln(a) for any r in Q and a>0, ie r is rational.

Homework Equations

ln(aⁿ)=nln(a)

The Attempt at a Solution

In a previous question I have already proved that ln(aⁿ)=nln(a), where n is a natural number. What I'm unsure about, is how is this any different? For 1), I'm not sure why you set b=aⁿ? Wouldn't you get ln((aⁿ)^1/n) = ln(a)? I'm not sure how this helps me find the solution.

Similarly for 2), I'm unsure how it is any different to proving that ln(aⁿ)=nln(a).

Any help would be great!

 

[Office_Shredder]

The obvious difference is that you proved a result $$m ln(a)=ln(a^m)$$ when m is a natural number, and now they want you to do it when m is not a natural number. The proof that you used for this will not work when m is not a natural number (unless you did something really clever) because at some point you assume that it was, probably in order to write a^m as a*a*a...*a a multiplied together m times (which doesn't make sense if m=1/2 for example)

If b=aⁿ, what does ln(b) and ln(b^1/n) become? Use it to do something cool

 

[pollytree]

The obvious difference is that you proved a result $$m ln(a)=ln(a^m)$$ when m is a natural number, and now they want you to do it when m is not a natural number. The proof that you used for this will not work when m is not a natural number (unless you did something really clever) because at some point you assume that it was, probably in order to write a^m as a*a*a...*a a multiplied together m times (which doesn't make sense if m=1/2 for example)

If b=aⁿ, what does ln(b) and ln(b^1/n) become? Use it to do something cool

Thanks for your help. I worked out part 1, but I'm still unsure on part 2. For proving the law for a natural number I did:
Let y=ln(a), for a>0,
e^y=a, and if we raise each side to the power n we get:
(e^y)ⁿ=aⁿ
ln(e^y*n)=ln(aⁿ)
y*n = ln(aⁿ), but y = ln(a)
n*ln(a) = ln(aⁿ)

So I'm still unsure how it's any different for any rational number?
Any help would be great :)

 

[fzero]

Use the definition of Q, namely that r =m/n, where m,n are in N.

 

[pollytree]

Ah I get it now! Thanks :)