Question

How do you write ${\log }_{2}1024=10$ in exponential form?

Answer 1

Hint: The logarithm functions are the inverse of the exponential functions. In exponential function, one term is raised to the power of another term, for example $a=x^y$ is an exponential function and the inverse of this function is $y={\log }_{x}a$ that is a logarithm function. Certain rules are followed by the logarithm functions that are called laws of the logarithm, using these laws we can write the function in a variety of ways.

Complete step-by-step answer:
In the given question, the logarithm function is written in the form of ${\log }_{a}b$ the base of the given function is 2. Using the concept mentioned above, we can solve the given equation and express it in exponential form.
We know that –
 $$\begin{gathered} if,{\log }_{n}x=a \\ \Rightarrow x=n^a \end{gathered}$$
So,
 $$\begin{gathered} {\log }_{2}1024=10 \\ \Rightarrow 1024=(2{)}^{10} \end{gathered}$$
Hence, the exponential form of ${\log }_{2}1024=10$ is $1024=2^{10}$ .
So, the correct answer is “ $2^{10}$ ”.

Note: The natural logarithm functions are denoted as $\ln a$ , they have the base of the logarithm function (x) as equal to e and can be written in log form as ${\log }_{e}a$ . $e$ is an irrational and transcendental mathematical constant, its value is nearly equal to $2.718281828459$ . There are three laws of the logarithm, two of the laws are for addition and subtraction of two or more logarithm functions and the third law is to convert logarithm functions to exponential functions. While applying the laws of the logarithm, the important condition is that the base of the logarithm functions involved should be the same in all the calculations. In the given question, we had to convert the logarithm function into the exponential function so we used the third law.