Exponents Linear algebra

Question

$x^y=z $

Proof that

$x^n/z=y$

I was calculating the cube root of $2$ by hand and when checking it out, I noticed its square is close to the value of logarithm of $3$ in base $2$. A little tweaking and I got the exact value for $n$ when $z$ is set at $3$. I wonder if there is detailed proof of such operations. Thanks.

n and z is fractional exponent where the index of the root just happens to equal the number we get when we make y the new exponent. — Adam Boit, Jan 30 '19 at 05:12

score 1 · Answer 1 · answered Jan 29 '19 at 21:45

1

Adding to abc..., note that: $$\log_b{x}=y \Rightarrow b^y=x$$

You're saying: $$2^{\frac{2}{3}}\approx\log_2{3}$$

If we break this into the components of a logarithm, we have: $b=2$, $y=\frac{2}{3}$, and $x=3$.

That would imply that $2^{\frac{2}{3}}=3$ by our definition of a logarithm.

Clearly, this is a contradiction though as $2^{\frac{2}{3}}\approx1.5874$. Therefore, this is just an interesting coincidence, not consequence of any "rule".

answered Jan 29 '19 at 21:45

Gnumbertester

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Thank you a lot. I have had this sudden craze in maths and I keep bumping on some numbers now and then. I'll share more. – Adam Boit Jan 29 '19 at 22:48
Glad I could help. I am going through the same thing, experimental math is fascinating. – Gnumbertester Jan 29 '19 at 23:15

score 0 · Answer 2 · answered Jan 29 '19 at 21:36

0

To me it's a coincidence.

$\log_2 3-2^{\frac{2}{3}}=0.002438551247043293297966695324491751631678920207371949352$

answered Jan 29 '19 at 21:36

abc...

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Exponents Linear algebra

2 Answers2