Question

express the following as a single log 1/3log27-2log1/3​

Answer 1

Step-by-step explanation:

a, b > 0# then:

#log ab = log a + log b#

and

#log (a/b) = log a - log b#

Hence:

#log a^n = n log a" "# for any integer #n > 0#

So we find:

#1/2 log 9 + 2 log 6 + 1/4 log 81 - log 12#

#=1/2 log 3^2 + 1/4 log 3^4 + log 6^2 - log 12#

#=1/2 (2 log 3) + 1/4 (4 log 3) + log (36/12)#

#= log 3 + log 3 + log 3#

#= 3 log 3#

Answer 2

$\large\sf\underline{Given}$

• $\sf\:\frac{1}{3}\:log(27)-2\:log\frac{1}{3}$

$\large\sf\underline{Solution}$

Let's simplify $\sf\:\frac{1}{3}$log(27)by moving $\sf\:\frac{1}{3}$inside the logarithm.

$\sf\:⟹\:log(27^{\frac{1}{3}})-2\:log\frac{1}{3}$

Rewrite 27 as $\sf\:3^{3}$

$\sf\:⟹\:log[(3^{3})^{\frac{1}{3}}]-2\:log\frac{1}{3}$

Applying power rule and multiplying exponents :

$\sf\:⟹\:log[(3^{3(\frac{1}{3})}]-2\:log\frac{1}{3}$

$\sf\:⟹\:log(3)-2\:log\frac{1}{3}$

$\sf\:⟹\:log(3)-log[(\frac{1}{3})^{2}]$

$\sf\:⟹\:log(3)-log\:\frac{1}{9}$

$\sf\:⟹\:log(3 \div \frac{1}{9})$

$\sf\:⟹\:log(3 \times \frac{9}{1})$

$\sf\:⟹\:log(27)$

So log (27) is the final answer.