Since you may be dealing with functions and relations, it would certainly be important to make out the best ways to deal with the identification of their domain and range. Exponential functions and logarithmic functions are known to be very closely tied to one another since a log or logarithm is just another way of writing exponential expressions.
Domain and range of Logarithmic Functions
Before we really begin, recall that the domain is the set of values for the input that may be entered for the expression and are also referred as the x values. The range set is similarly the set of values for y or the probable outcome.
Now, consider that an exponential function is defined as $y=b^{x}$; for any real number i.e. $x $ and constant $b>0, b\ne 1.$ also, where,
- The domain for y is $(-\infty , \infty )$
- And the range for y is $(0 , \infty )$
You may be aware of the fact that the inverse of the given exponential function $y=b^{x}$ can be written as the logarithmic function $y=\log _{b}(x). $ Also consider that
- The domain for $y=\log _{b}(x)$ will be the range of $y=b^{x}:(0,\infty ).$
- The range for $y=\log _{b}(x)$ will be the domain of $y=b^{x}:(-\infty , \infty ).$
The transformations of the parent function $y=\log _{b}(x)$ tend to be behaving similarly to other functions. Just like the case with other parent functions, major four types of transformations could be applied to the parent function without the loss of shape. These include stretches, shifts, reflections, and compressions.
If the graphs of exponential functions are considered, various transformations can change or deviate the range of $y=b^{x}$. Similarly, these transformations when applied to the logarithmic function $y=\log _{b}(x)$ can result in the change in domain. It is therefore important to keep in mind that the domain could consist of only positive values and real numbers. This implies that the argument of logarithmic functions is likely to be greater than zero.
Consider the example where $f(x)=\log _{4}(2x-3)$ has been defined for a value $x$ such that the argument $(2x-3)$ would be greater than zero. In order to find the domain for the same, we would be required to set up the inequality and solve for $x$.
$2x-3>0$: Show up the argument greater than zero.
$2x>3$ : Add 3.
$x>1.5$ : Divide by 2.
In the interval notation, the domain for $f(x)=\log _{4}(2x-3)$ would be $(1.5,\infty ). $
Considering the above illustration, it would be easier to make out the domain for any given logarithmic functions.
