Question

How do you solve $ {{\log }_{2}}\left( 2x \right)={{\log }_{2}}100 $ ?

Answer 1

Hint: From the question given we have been asked to solve $ {{\log }_{2}}\left( 2x \right)={{\log }_{2}}100 $ . We can clearly observe that the given question is in the form of logarithms. We can solve the logarithmic equations by using some simple transformations.

Complete step by step answer:
From the question, it has been given that $ {{\log }_{2}}\left( 2x \right)={{\log }_{2}}100 $
We can clearly observe that, in the above-given equation, the bases of both the right-hand side of the equation and the left-hand side of the equation are equal.
In logarithms, if bases of both right hand side of the equation and left hand side of the equation are equal, we have one formula if $ {{\log }_{a}}b={{\log }_{a}}c $ , then $ b=c $
Now, we have to apply the above-shown formula to solve the given logarithmic equation in the equation.
By applying the above written logarithmic formula to the given logarithmic question, we get $ {{\log }_{2}}\left( 2x \right)={{\log }_{2}}100 $
As the bases are equal, we can essentially cancel them out and be left with $ 2x=100 $
Now, we have to do furthermore simplification to get the simplified answer.
Divide both sides of the equation by $ 2 $ to get the simplification.
On dividing both sides of the equation by $ 2 $ , we get
 $ \dfrac{2x}{2}=\dfrac{100}{2} $
 $ \Rightarrow x=50 $
Hence, the given logarithmic equation is simplified.

Note:
We should be well aware of the logarithms. We should be well known about the properties of the logarithms and also we should be well known about the basic formulae of logarithms. We should also know which basic formula is to be used to solve the given question, to do that we have to understand the question very clearly. There are many other similar formulae like $ \log ab=\log a+\log b $ and $ \log \left( \dfrac{a}{b} \right)=\log a-\log b $ many more.