Question

What is the domain and range of $y = \log {2^x}$ ?

Answer 1

Here we are going to see how to find the domain and range of the logarithmic function. The domain of the logarithmic function is real numbers greater than zero, and the range is real numbers. To determine the domain and range of a function, first determine the set of values for which the function is defined and then determine the set of values that result from these.

Complete step-by-step answer:
Logarithmic functions are the inverse of the exponential function. The inverse of the exponential function $y = {a^x}$ is $x = {a^y}$. The logarithmic function $y = {\log _a}x$ is defined to be equivalent to the exponential equation $x = {a^y}$. $y = {\log _a}x$only under the following conditions $x = {a^y}$,$a > 0$ , and $a \ne 1$it is called logarithmic function with base $a$.
The domain of the logarithmic function is real numbers greater than zero, and the range is real numbers. The graph $y = {\log _a}x$ is symmetrical to the graph $y = {a^x}$with respect to the line $y = x$ the relationship is true for any function and its inverse.
Here are some useful properties of logarithms, which all follow from identities involving exponents and the definition of the logarithm. Remember $a > 0$, and $x > 0$
That is $\log {m^n} = n\log m$
From our given function $y = \log {2^x}$
We can apply this property we get,
$\Rightarrow y = x\log 2$
Now, this becomes a linear equation,
We know that $\log 2 \approx 0.301$ then we get,
$\Rightarrow y = 0.301x$
From this, we came to believe that the values of both $x$ and $y$ belong to real numbers that are $x \in \mathbb{R}$ and $y \in \mathbb{R}$ .

Note: A function is a relationship between the x and y values, where each $x$ value or input has only one $y$ value or output.
Domain: all $x$ values or inputs that have an output of real $y$ values.
Range: the $y$ values or outputs of a function.