Logarithm Proofs

1. Oct 6, 2006

thomasrules

Can't start:

$$(log_{a}b)(log_{b}a) =1$$

2. Oct 6, 2006

arildno

Are you to prove that one?
Note that:
$$a^{\log_{a}(b)*\log_{b}(a)}=(a^{\log_{a}(b)})^{\log_{b}(a)}$$
Can you continue?

3. Oct 6, 2006

thomasrules

how'd u get that

4. Oct 6, 2006

thomasrules

5. Oct 6, 2006

It should be $$a^{\log_{a}(b)} = b$$. My fault.

6. Oct 6, 2006

thomasrules

i'm sorry but i don't even understand that step from the original to that

7. Oct 6, 2006

You get $$a^{\log_{a}(b)*\log_{b}(a)} = a^{1}$$ therefore $$(log_{a}b)(log_{b}a) =1$$. Basically, you start with the base $$a$$ and raise it to the respective powers on the left and right hand side of the equation. You could have used $$b$$ as the base instead.

Also $$a^{\log_{a}(b)} = b$$. Look at an example. $$10^{\log_{10}(100)} = 10^{2} = 100 = b$$

Last edited: Oct 6, 2006
8. Oct 6, 2006

thomasrules

so then:

$$log_{a}a=(log_{a}b)(log_{b}a)$$

???

9. Oct 6, 2006

$$\log_{a}a = 1$$ is true. But that is not how I showed that $$(\log_{a}b)(\log_{b}a) =1$$. We have $$a^{\log_{a}(b)*\log_{b}(a)} = a^{1}$$. Since the bases of both sides of the equation is $$a$$, we can equate their exponents to each other. That means $$(\log_{a}b)(\log_{b}a) =1$$.

10. Oct 6, 2006

thomasrules

but that's not the proof.....

11. Oct 6, 2006

arildno

By the normal rule for a product in the exponent.

12. Oct 7, 2006

lkh1986

We can change the base for the second term, which gives
log (base b) a = log (base a ) a / log (base a) b

From there, we can cancel out the term log (base a) b and only log (base a ) a remains, which gives the answer of 1.

13. Oct 8, 2006

The Bob

Try looking at the 2nd post.

Here is the proof that you want. As arildno said, continue this and you will find that $$(log_{a}b)(log_{b}a) =1$$, which is what you said you wanted. All you need to use is the laws of indices, the idea of how logs work and you are done, no joke.

All the best,