Question

How do you write ${\log _4}16 = 2$ in exponential form?

Answer 1

Hint: The inverse of the exponential functions are called logarithm 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 called the laws of the logarithm are followed by the logarithm functions, we can write the function in a variety of ways using these laws. In the given question, the logarithm function is written in the form of ${\log _a}b$ the base of the given function is 2. We can solve the given equation and express it in exponential form by using the above information.

Complete step-by-step solution:
We know that –
$
  if,\,{\log _n}x = a \\
   \Rightarrow x = {n^a} \\
 $
Here the given data is ${\log _4}16 = 2$ which need to be written in exponential form
So,
$
  {\log _4}16 = 2 \\
   \Rightarrow 16 = {(2)^4} \\
 $
Hence, the exponential form of ${\log _4}16 = 2$ is $16 = {2^4}$ .


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. The base of the logarithm functions involved should be the same in all the calculations while applying the laws of the logarithm. In the given question, we used the third law to convert the logarithmic function into the exponential function.