Question

evaluate the given question​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{\log_{c^2}(a^b)\cdot\,\log_{a^2}(b^c)\cdot\,\log_{b^4}(c^a)}$

$\tt{=b\log_{c^2}(a)\cdot\,c\log_{a^2}(b)\cdot\,a\log_{b^4}(c)}$

$\tt{=\dfrac{b}{2}\log_{c}(a)\cdot\,\dfrac{c}{2}\log_{a}(b)\cdot\,\dfrac{a}{4}\log_{b}(c)}$

$\tt{=\dfrac{b}{2}\dfrac{\ln(a)}{\ln(c)}\cdot\,\dfrac{c}{2}\dfrac{\ln(b)}{\ln(a)}\cdot\,\dfrac{a}{4}\dfrac{\ln(c)}{\ln(b)}}$

$\tt{=\dfrac{b}{2}\cdot\,\dfrac{c}{2}\cdot\,\dfrac{a}{4}}$

$\tt{=\dfrac{abc}{16}}$

Answer 2

Given Question : -

Evaluate :-

$\rm :\longmapsto\:{\log_{c^2}(a^b)\cdot\,\log_{a^2}(b^c)\cdot\,\log_{b^4}(c^a)}$

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:{\log_{c^2}(a^b)\cdot\,\log_{a^2}(b^c)\cdot\,\log_{b^4}(c^a)}$

We know,

$\rm :\longmapsto\:\boxed{ \tt{ \: log_{ {x}^{y} }( {z}^{w} ) = \frac{w}{y} log_{x}(z) \: }}$

So, using this identity, we get

$\rm \:  =  \:\dfrac{b}{2} log_{c}(a) \dfrac{c}{2} log_{a}(b) \dfrac{a}{4} log_{b}(c)$

$\rm \:  =  \:\dfrac{abc}{16} log_{c}(a) \times log_{a}(b) \times log_{b}(c)$

We know,

$\rm :\longmapsto\:\boxed{ \tt{ \: log_{x}(y) = \frac{logy}{logx} \: }}$

So, using this, we get

$\rm \:  =  \:\dfrac{abc}{16} \times \dfrac{loga}{logc} \times \dfrac{logb}{loga} \times \dfrac{logc}{logb}$

$\rm \:  =  \:\dfrac{abc}{16}$

More to know :-

$\boxed{ \tt{ \: log_{x}(x) = 1 \: }}$

$\boxed{ \tt{ \: log_{x}( {x}^{y} ) =y \: }}$

$\boxed{ \tt{ \: log_{ {x}^{z} }( {x}^{y} ) = \frac{y}{z} \: }}$

$\boxed{ \tt{ \: {e}^{logx} \: = \: x \: }}$

$\boxed{ \tt{ \: {e}^{y \: logx} \: = \: {x}^{y} \: }}$

$\boxed{ \tt{ \: {a}^{ log_{a}(x) } = x \: }}$

$\boxed{ \tt{ \: {a}^{ ylog_{a}(x) } = {x}^{y} \: }}$