Question

If log2(x)= a and log3(y)= b then write 72^a in term or x and y.​

Answer 1

Question :

If $\sf log_{2}(x) = a \: and \: \: log_{3}(y) = b$

Then write $\sf\:72{}^{a}$ in term of x and y .

Theory :

If $log_{b}(a) = x$ ,in exponent form : $\implies \: b {}^{x} = a$

Solution :

Given that :$\sf\:\log_{2}(x)=a$ and $\sf\:\log_{3}(y)=b$

Now , $\sf\:\log_{2}(x)=a$

$\sf\:\implies\:2{}^{a}=x..(1)$

and $\sf\:\log_{3}(y)=b$

$\sf\:\implies\:3{}^{b}=y..(2)$

Prime Factorization of 72

$\sf72 = 2 \times 2 \times 2 \times \times 3 \times 3$

Hence, $\sf72 {}^{a} = (2 \times 2 \times 2 \times3 \times 3) {}^{a}$

$\sf = (2 {}^{3} \times 3 {}^{2} ) {}^{a}$

$\sf = {(2 {}^{a}) }^{3} \times {(3 {}^{b} )}^{2}$

From equation (1) &(2)

$\sf = x {}^{3}y {}^{2}$

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More About Logarithm:

The Logarithm function is defined as

$f(x) = log_{b}(x)$

where b > 0 and b ≠ 1 and also x >0, reads as log base b of x.

⇒Generally we use the base 10 i.e $\log_{10}$