Question

❥Prove that attachment.​

Answer 1

Proof :

$\sf log_{y}( {x}^{2} ). log_{z}( {y}^{3} ) .log_{x}( {z}^{5} )log$

Use Logarithms rule,

$\boxed{\sf log_{a} {m}^{n} = n log_{a}m}$

Therefore,

$\sf = 2 log_{y}x \times 3 log_{z}y \times 5 log_{xz = 2 log}$

$=\sf 30 log_{y}x . log_{z}y. log_{x}z=30log$

We know,

$\boxed{ \sf log_{b}m = \frac{ log_{a}m }{ log_{a}b}}$

Therefore,

$\sf= 30 \times \frac{logx}{logy} \times \frac{logy}{logz} \times \frac{logz}{logx}$

$\sf = 30=30 = RHS$

Hence proved !!.

Answer 2

To Prove,

$\bull \sf \: log_yx {}^{2} .log_zy {}^{3} .log_xz {}^{5} = 30$

Proof,

LHS

$\sf \: log_yx {}^{2} .log_zy {}^{3} .log_xz {}^{5}$

[ Using logarithm rule ]

$\blue\bigstar \sf \: \log_am {}^{n} = nlog_am$

$\dashrightarrow \sf \: 2log_yx \: . \: 3log_zy \: . \: 5log_xz\: \pink\bigstar$

[ We know that ]

$\sf \red \bigstar \: log_bm= \dfrac{log_am}{log_ab}$

$\dashrightarrow \sf \: 2 \cancel\dfrac{log \: x}{log \:y} \times 3 \cancel\dfrac{log \: y}{log \: z} \times 5 \cancel\dfrac{log \: z}{log \: x}$

$\dashrightarrow \sf \: 2 \times 3 \times 5$

$\dashrightarrow \sf \: 30 \: \green \bigstar$

RHS

Hence Proved !!