findout whether log8 32 is rational (or) irrational? justify. why
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Answer:
It is Rational.
We assume that \log 2log2 is a rational number.
So, We can write \log 2log2 in form of p/q where p and q are integers and q is non-zero.
\log_{10} 2=\frac{p}{q}log
10
2=
q
p
We know, \log_b a=x\Rightarrow a=b^xlog
b
a=x⇒a=b
x
2=10^{\frac{p}{q}}2=10
q
p
2=(2\times 5)^{\frac{p}{q}}2=(2×5)
q
p
2^q=(2\times 5)^{p}2
q
=(2×5)
p
2^{q-p}=(5)^{p}2
q−p
=(5)
p
Where, q-p is an integer greater than zero.
Now, it can be seen that the L.H.S. is even and the R.H.S. is odd.
So, there is contradiction.
As \log 2log2 is an irrational number.
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