prove that the following are irrational numbers
1) tan 5
2) log 6 to the base 7
Hello, fxs12!
Here's the second one . . .
$\displaystyle \text{Prove that the following are irrational numbers: }$
. . $\displaystyle (1)\;\tan 5 \qquad\qquad (2)\;\log_76$
Suppose $\displaystyle \log_76$ is rational.
Then: .$\displaystyle \log_76 \:=\:\dfrac{n}{d}\:\text{ for some integers }n,d$
. . And we have: .$\displaystyle 7^{\frac{n}{d}} \:=\:6$
Raise both sides to the power $\displaystyle \,d\!:\;\;7^n \:=\:6^d
$
. . And we have: .$\displaystyle 7^n \;=\;2^d\cdot3^d$
We have a number that has two distinct prime factorizations.
This is contradiction of the Fundamental Theorem of Arithmetic.
Therefore: .$\displaystyle \log_76$ is irrational.