# Functions Exercises | Logarithmic and Exponential Functions

### Exercises for logarithmic and exponential functions involve tasks like graphing, solving equations, and applying these functions to real-world problems. These exercises help build a solid understanding of how logarithmic and exponential functions work and how they are used in various contexts.

The exercise

1- We consider the function g defined on the interval ]0, +∞[ as follows: g(x)= -xlnx + e

a - Analyze the variations of the function g and create a table of its variations.

b- Calculate g(e), then deduce the sign of g(x) over the interval ]0, +∞[.

2- We define the function f on the interval ]0, +∞[  as follows: f(x)= 1/2 (xlnx)² + ex  — e    and (Cf) is the graph of the function f in the coordinate plane associated with the orthogonal and homogeneous vectors (o;   i乛,j乛)  in which:                     ||乛i  ||=2cm  and  ||乛j  ||=4cm

a- Calculate the limit of f at 𝑫f and then interpret the result graphically.

b- Show that for any real number ]0, +∞[: f'(x)=(1/x)

c- Demonstrate that g is strictly decreasing on ]0, 1/e[ And strictly increasing on ]1/e, +∞ Then create a table of its variations.

3-

a- Calculate f(1), then show that f(x) = 0 has a unique solution in the interval ]0.11, 0.12 [.

b- Show that y=e(x-1) is the equation of the tangent line (T) to the curve (Cf) at the point with the abscissa 1.

c- Analyze the position of (T) with respect to (Cf).

d- Create (T) and (Cf).

4- We consider the function h defined by ]0,+∞[ by h(x)=| f(x)|, (Ch) graphically represent them in the previous coordinate system.

a-  Explain how to construct (Ch) from the curve (Cf) Then create (Ch)

5-

a- We put
Prove by integration by division that I=e²+1

b- We put

Prove by integration by division that J=4e²2I

c- Calculate, in cm, the area S, the area S of the region bounded by the curve (C) and the lines with equations:
x=1, x=e², and y=0

d ه   || ||