Question

How do you solve $ 6{\log _3}\left( {0.5x} \right) = 11 $

Answer 1

Hint: For solving this question, firstly divide $ 6 $ both sides so that the constants come to one side and then make the base both sides so add base $ 3 $ . Now take inverse both sides, logarithm would be eliminated. Hence, we will find the value of x.

Complete step by step solution:
In the question, we are given the expression $ 6{\log _3}\left( {0.5x} \right) = 11 $ and we have to find the value of $ x $
For solving this expression, firstly dividing $ 6 $ both sides
 $ {\log _3}\left( {0.5x} \right) = \dfrac{{11}}{6} $
Now, making the exponent of a base $ 3 $ on both sides. So, on taking the inverse function the log will cancel out.
 $\Rightarrow {3^{{{\log }_3}\left( {0.5x} \right)}} = {3^{\dfrac{{11}}{6}}} $
Now, take the inverse on the left-hand side.
 $\Rightarrow \left( {0.5x} \right) = {3^{\dfrac{{11}}{6}}} $
Dividing both sides by $ 0.5 = \dfrac{5}{{10}} = \dfrac{1}{2} $
 $\Rightarrow x = 2 \times {3^{\dfrac{{11}}{6}}} $
 $\Rightarrow x \cong 2 \times 7.5 \cong 15 $
Hence, the solution of the question is $ 15 $ approx.
So, the correct answer is “ $ 15 $ approx”.

Note: Be careful on which step should be taken next. The steps should be taken according to the question. On seeing the logarithm, try to remove it first because it will make the question look easier. The basic calculations should be done properly.