Proof of logarithmic properties.

[pollytree]

1. The problem statement, all variables and given/known data

There are two log properties that I have to prove:
1) Explain why ln(b1/n)=(1/n)ln(b) for b>0, set b=an

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

2. Relevant equations

ln(an)=nln(a)

3. 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 am 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(b1/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 am 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(b1/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,
ey=a, and if we raise each side to the power n we get:
(ey)n=an
ln(ey*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 :)