Question

How do you write ${\log _2}(x) = 5$ in exponential form ?

Answer 1

Hint: In this question, we need to find out the exponential form of a given logarithmic function. We know that the logarithm functions are the inverse of exponential functions. In exponential function, one term is raised to the power of another term, i.e. of the form $x = {b^y}$ is an exponential function and the inverse of this function is $y = {\log _b}x$ which is a logarithmic function. Since in this question we are given a logarithmic function, we convert it using this rule explained.

Complete step by step solution:
Given the logarithmic function ${\log _2}(x) = 5$ …… (1)
We are asked to convert the logarithmic function in the equation (1) into an exponential function.
We know that logarithm is the inverse of exponential function. Note that logarithm form and index (exponential) form are interchangeable.
i.e. if ${b^y} = x$, then we have ${\log _b}x = y$ …… (2)
Where log denotes the logarithmic function.
Here x is an argument of logarithm function which is always positive.
And b is called the base of the logarithm function.
Using the concept mentioned above, we can solve the given equation and express it in exponential form.
Now consider the given function given in the equation (1).
Now compare it with the general form given in the equation (2).
We have here $b = 2$ and $y = 5$
Thus we get,
${\log _2}(x) = 5$
$ \Rightarrow {2^5} = x$

Hence the exponential form of ${\log _2}(x) = 5$ is given by ${2^5} = x$.

Note :
If the question has the word log or $\ln $, it represents the given function as logarithmic function. Note that we have two types of logarithmic function.
One is a common logarithmic function which is represented as a log and its base is 10.
The other one is a natural logarithmic function represented as $\ln $ and its base is $e$.
We must know the basic properties of logarithmic functions and note that these properties hold for both log and $\ln $ functions.
Some properties of logarithmic functions are given below.
(1) $\log (x \cdot y) = \log x + \log y$
(2) $\log \left( {\dfrac{x}{y}} \right) = \log x - \log y$
(3) $\log {x^n} = n\log x$
(4) $\log 1 = 0$
(5) ${\log _e}e = 1$