Transcendental function - Practice Questions & MCQ

Transcendental functions: the functions which are not algebraic are called transcendental functions. Exponential, logarithmic, trigonometric and inverse trigonometric functions are transcendental functions.

Exponential Function: function $f(x)$ such that $f(x)=a^x$ is known as an exponential function.

$
\begin{aligned}
& \text { base: } \quad a>0, a \neq 1 \\
& \text { domain : } x \in \mathbb{R} \\
& \text { range : } f(x)>0
\end{aligned}
$
 

Property: If $\mathrm{a}^{\mathrm{x}}=\mathrm{a}^{\mathrm{y}}$, then $\mathrm{x}=\mathrm{y}$
Logarithmic function: function $\mathrm{f}(\mathrm{x})$ such that $f(x)=\log _a(x)$ is called logarithmic function
base: $\quad \mathrm{a}>0, \mathrm{a} \neq 1$
domain : $x>0$
range : $\mathrm{f}(\mathrm{x}) \in \mathbb{R}$
         

                    If a > 1                                                                               If 0 < a < 1

Properties of Logarithmic Function

1. $\log _e(a b)=\log _e a+\log _e b$
2. $\log _{\mathrm{e}}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)=\log _{\mathrm{e}} \mathrm{a}-\log _{\mathrm{e}} \mathrm{b}$
3. $\log _e \mathrm{a}^{\mathrm{m}}=\mathrm{m} \log _{\mathrm{e}} \mathrm{a}$
4. $\log _{\mathrm{a}} \mathrm{a}=1$
5. $\log _{\mathrm{b}^{\mathrm{m}}} \mathrm{a}=\frac{1}{\mathrm{~m}} \log _{\mathrm{b}} \mathrm{a}$
6. $\log _{\mathrm{b}} \mathrm{a}=\frac{1}{\log _{\mathrm{a}} \mathrm{b}}$
7. $\log _{\mathrm{b}} \mathrm{a}=\frac{\log _{\mathrm{m}} \mathrm{a}}{\log _{\mathrm{m}} \mathrm{b}}$
8. $\mathrm{a}^{\log _{\mathrm{a}} \mathrm{m}}=\mathrm{m}$
9. $\mathrm{a}^{\log _c \mathrm{~b}}=\mathrm{b}^{\log _{\mathrm{c}} \mathrm{a}}$
10. $\log _{\mathrm{m}} \mathrm{a}=\mathrm{b} \Rightarrow \mathrm{a}=\mathrm{m}^{\mathrm{b}}$