Question

Express in terms of log 2 and log 3 log root 54^1/2 * log 243^1/3

Answer 1

i guess the question is wrong

it should be

log root 64^1/2 * log 243^1/3

then the answer is 5 / 2 log 3 * log 2

Answer 2

The simplified form of the given expression is $\dfrac{5}{4}(\log 3)^2+\dfrac{5}{12} \log 2\log 3$.

Step-by-step explanation:

Consider the given expression is

$\log \sqrt{54^{1/2}}\times \log (243^{1/3})$

It can be rewritten as

$\log (54^{1/2}})^{1/2}\times \log (243^{1/3})$                  $[\because \sqrt[n]{x}=x^{1/n}]$

Using the properties of exponent and log we get

$\log ((3^3\cdot 2)^{1/4})\times \log ((3^5)^{1/3})$

$\log ((3^{3/4}\cdot 2^{1/4})\times \log (3)^{5/3}$                   $[\because (ab)^m=a^mb^m]$

$(\log 3^{3/4}+ \log 2^{1/4})\times \log (3)^{5/3}$                   $[\because \log ab=\log a+\log b]$

$(\dfrac{3}{4}\log 3+\dfrac{1}{4} \log 2)\times \dfrac{5}{3}\log (3)$                  $[\because \log a^b=b\log a]$

$\dfrac{5}{4}(\log 3)^2+\dfrac{5}{12} \log 2\log 3$

Therefore, the simplified form of the given expression is $\dfrac{5}{4}(\log 3)^2+\dfrac{5}{12} \log 2\log 3$.

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