Question

2log 2 +log 5-1/2 log 36-log 1/30​

Answer 1

Step-by-step explanation:

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Answer 2

Question:

Evaluate:

$\displaystyle{\sf\:2\:\log\:(\:2\:)\:+\:\log\:(\:5\:)\:-\:\dfrac{1}{2}\:\log\:(\:36\:)\:-\:\log\:\left(\dfrac{1}{30}\:\right)}$

Answer:

$\displaystyle{\boxed{\red{\sf\:2\:\log_b\:(\:2\:)\:+\:\log_b\:(\:5\:)\:-\:\dfrac{1}{2}\:\log_b\:(\:36\:)\:-\:\log_b\:\left(\dfrac{1}{30}\:\right)\:=\:2\:\log_b\:(\:10\:)\:}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:2\:\log\:(\:2\:)\:+\:\log\:(\:5\:)\:-\:\dfrac{1}{2}\:\log\:(\:36\:)\:-\:\log\:\left(\dfrac{1}{30}\:\right)}$

We have to evaluate the given expression.

Now,

$\displaystyle{\sf\:2\:\log\:(\:2\:)\:+\:\log\:(\:5\:)\:-\:\dfrac{1}{2}\:\log\:(\:36\:)\:-\:\log\:\left(\dfrac{1}{30}\:\right)}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:\log_b\:(\:a^k\:)\:=\:k\:\log_b\:(\:a\:)}}}$

$\displaystyle{\implies\sf\:\log\:(\:2^2\:)\:+\:\log\:(\:5\:)\:-\:\log\:(\:36^{\frac{1}{2}}\:)\:-\:\log\:\left(\:\dfrac{1}{30}\:\right)}$

$\displaystyle{\implies\sf\:\log\:(\:4\:)\:+\:\log\:(\:5\:)\:-\:\log\:(\:\sqrt{36}\:)\:-\:\log\:\left(\:\dfrac{1}{30}\:\right)}$

$\displaystyle{\implies\sf\:\log\:(\:4\:)\:+\:\log\:(\:5\:)\:-\:\log\:(\:6\:)\:-\:\log\:\left(\:\dfrac{1}{30}\:\right)}$

We know that,

$\displaystyle{\boxed{\blue{\sf\:\log_b\:(\:x\:)\:+\:\log_b\:(\:y\:)\:=\:\log_b\:(\:xy\:)}}}$

$\displaystyle{\implies\sf\:\log\:(\:4\:\times\:5\:)\:-\:\log\:(\:6\:)\:-\:\log\:\left(\:\dfrac{1}{30}\:\right)}$

$\displaystyle{\implies\sf\:\log\:(\:20\:)\:-\:\log\:(\:6\:)\:-\:\log\:\left(\:\dfrac{1}{30}\:\right)}$

We know that,

$\displaystyle{\boxed{\green{\sf\:\log_b\:(\:x\:)\:-\:\log_b\:(\:y\:)\:=\:\log_b\:\left(\:\dfrac{x}{y}\:\right)}}}$

$\displaystyle{\implies\sf\:\log\:\left(\:\dfrac{\cancel{20}}{\cancel{6}}\:\right)\:-\:\log\:\left(\:\dfrac{1}{30}\:\right)}$

$\displaystyle{\implies\sf\:\log\:\left(\:\dfrac{10}{3}\:\right)\:-\:\log\:\left(\:\dfrac{1}{30}\:\right)}$

$\displaystyle{\implies\sf\:\log\:\left(\:\dfrac{\dfrac{10}{3}}{\dfrac{1}{30}}\:\right)}$

$\displaystyle{\implies\sf\:\log\:\left(\:\dfrac{10}{\cancel{3}}\:\times\:\cancel{30}\:\right)}$

$\displaystyle{\implies\sf\:\log\:(\:10\:\times\:10\:)}$

$\displaystyle{\implies\sf\:\log\:(\:10^2\:)}$

$\displaystyle{\implies\sf\:2\:\log\:(\:10\:)}$

Taking the base of logarithm as b, we get,

$\displaystyle{\implies\sf\:2\:\log_b\:(\:10\:)}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:2\:\log_b\:(\:2\:)\:+\:\log_b\:(\:5\:)\:-\:\dfrac{1}{2}\:\log_b\:(\:36\:)\:-\:\log_b\:\left(\dfrac{1}{30}\:\right)\:=\:2\:\log_b\:(\:10\:)\:}}}}$