###

**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 + ****e****x — **** e **and (C

*f*) 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 𝑫*f* 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 (C

*f*) at the point with the abscissa 1.

**c-** Analyze the position of (T) with respect to (C*f*).

**d-** Create (T) and (C*f*).

**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 (C*h)* from the curve (C*f*) Then create (C*h)*

**5-**

**a-**We put

**I=e²+1**

**b-**We put

**J=4e²**

**—****2I**

**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**

|| ||