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If-x-2-p-and-y-4-q-then-prove-that-log-2-x-3-y-3p-2q-




Question Number 193360 by MATHEMATICSAM last updated on 11/Jun/23
If x = 2^p  and y = 4^q  then prove that  log_2 (x^3 y) = 3p + 2q
$$\mathrm{If}\:{x}\:=\:\mathrm{2}^{{p}} \:\mathrm{and}\:{y}\:=\:\mathrm{4}^{{q}} \:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{log}_{\mathrm{2}} \left({x}^{\mathrm{3}} {y}\right)\:=\:\mathrm{3}{p}\:+\:\mathrm{2}{q} \\ $$
Answered by AST last updated on 11/Jun/23
log_2 (2^(3p) 2^(2q) )=log_2 (2^(3p+2q) )=3p+2q
$${log}_{\mathrm{2}} \left(\mathrm{2}^{\mathrm{3}{p}} \mathrm{2}^{\mathrm{2}{q}} \right)={log}_{\mathrm{2}} \left(\mathrm{2}^{\mathrm{3}{p}+\mathrm{2}{q}} \right)=\mathrm{3}{p}+\mathrm{2}{q} \\ $$
Answered by Frix last updated on 11/Jun/23
(b^m )^n =b^(mn)   b^m b^n =b^(m+n)   log_b  b^a  =a    x^3 y=2^(3p) 4^q =2^(3p) (2^2 )^q =2^(3p) 2^(2q) =2^(3p+2q)   log_2  2^(3p+2q)  =3p+2q
$$\left({b}^{{m}} \right)^{{n}} ={b}^{{mn}} \\ $$$${b}^{{m}} {b}^{{n}} ={b}^{{m}+{n}} \\ $$$$\mathrm{log}_{{b}} \:{b}^{{a}} \:={a} \\ $$$$ \\ $$$${x}^{\mathrm{3}} {y}=\mathrm{2}^{\mathrm{3}{p}} \mathrm{4}^{{q}} =\mathrm{2}^{\mathrm{3}{p}} \left(\mathrm{2}^{\mathrm{2}} \right)^{{q}} =\mathrm{2}^{\mathrm{3}{p}} \mathrm{2}^{\mathrm{2}{q}} =\mathrm{2}^{\mathrm{3}{p}+\mathrm{2}{q}} \\ $$$$\mathrm{log}_{\mathrm{2}} \:\mathrm{2}^{\mathrm{3}{p}+\mathrm{2}{q}} \:=\mathrm{3}{p}+\mathrm{2}{q} \\ $$

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