Question

How do you simplify ${{\log }_{2}}\left( {{2}^{x}} \right)$

Answer 1

Hint: Logarithmic function is an inverse function of the exponential function. An exponential function is given as $y={{a}^{x}}$, when ‘a’ is the base and ‘x’ is the exponent. Use the properties of logarithmic functions and simplify the given expression.

Complete step-by-step solution:
The given expression is a logarithmic function. Let us first understand what is a logarithmic function and then understand its properties.
Logarithmic function is an inverse function of the exponential function. An exponential function is given as $y={{a}^{x}}$, when ‘a’ is the base and ‘x’ is the exponent. We read this as y is equal to a raised to the power x.
A logarithmic function for the above exponential function is given as $x={{\log }_{a}}y$. We read this equation as x is equal to log to the base ‘a’ of y.
From the above logarithmic equation we understand that the function tells us about the exponent in the corresponding exponential function.
In the expression given in the question, the base of the logarithm is 2.
One of the property of logarithm functions say that ${{\log }_{a}}{{y}^{b}}=b{{\log }_{a}}y$.
Therefore, we can write ${{\log }_{2}}\left( {{2}^{x}} \right)$ as ${{\log }_{2}}\left( {{2}^{x}} \right)=x.{{\log }_{2}}\left( 2 \right)$ …. (i)
There is another property that says that ${{\log }_{a}}(a)=1$.
Then this means that ${{\log }_{2}}(2)=1$
Substitute this value in equation (i).
This gives us that ${{\log }_{2}}\left( {{2}^{x}} \right)=x(1)=x$
Hence, we simplified the given expression and we found that ${{\log }_{2}}\left( {{2}^{x}} \right)=x$

Note: In this given question we simplified for the logarithm function to the base 2. And from this question we can general an equation for any base that ${{\log }_{a}}\left( {{a}^{x}} \right)=x$.Now, the good thing is that we can directly use this property whenever we come across such expression.