Logarithm: senior math exercises corrected in PDF.

Logarithms with corrected senior math exercises involve several interesting properties as well as functions.

These corrected exercises on logarithms involve the following concepts:

  1. definition of the logarithm;
  2. functional equations;
  3. algebraic formulas on logarithms;
  4. limits and logarithm functions.

Exercise #1:

Solve the following equations:


Exercise #2:

Solve the following equations:


Exercise #3:

Simplify the writing of the following numbers:


Exercise #4:

After having specified the set of definition of the solutions of the equation, solve it.


Exercise #5:
Let f be the function defined on %5D0;+\infty%5B by: f(x)=2x-1+ln\,(\,\frac{x}{x+1}\,\,).
We note C_f its graphic representation in an orthonormal reference frame \,(\,O;\vec{i},\vec{j}\,\,) of the plane (graphic unit: 2 cm).
1. Study the limit of f at 0. Interpret this result graphically.
2. a. Study the limit of f in +\infty.
b. Show that the line \Delta with equation y=2x-1 is an asymptote to C_f in +\infty.
Study the position of C_f in relation to \Delta.
3. Study the variations of f. Draw up its table of variations.
4. Prove that the equation f(x) = 0 has a unique solution \alpha in the interval %5D0;+\infty%5B and determine a frame of \alpha of magnitude 10^{-2}.
5. Draw the line \Delta and the curve C_f.

Exercise #6:

Also useful for the baccalaureate… in Chemistry!
In chemistry, we know that the pH of a solution allows us to express its acidic or basic character.
This number is a decimal between 1 and 14 so that :
● If pH < 7, then the solution is said to be acidic.
● If pH > 7, then the solution is said to be basic.
● If pH = 7, it is said to be neutral.

ph of a solution and logarithm
It is then known that the pH is associated with the relation pH\,=\,-\,log%5BH_3O^{+}%5Dwhere %5BH_3O^+%5D is the concentration of ions H_3O^+, expressed in mol/L.

1. A solution has a concentration of ions H_3O^+ equal to 5\times  \,10^{-9}.
What is its pH? What can we say about a solution whose ion concentration is H_3O^+?
equal to 0.1 ?
2. What is the ion concentration H_3O^+ of a neutral solution?
3. If you increase the concentration of H_3O^+ ions in a solution, do you decrease or increase the pH of the solution?
4. What must be done to a solution to increment or decrement its pH?

Vocabulary : To increment is to add 1. So decrementing is… ?

Exercise #7:
f is the function defined on \mathbb{R} by :
C is its curve in an orthogonal reference frame \,(\,O;\vec{i},\vec{j}\,\,).

1. a. Determine the limit of ln(1+e^{2x}) in -\infty.
b. Deduce the existence of an oblique asymptote \Delta of which we will specify an equation.
c. Show that for any real x :
d. Determine the limit of f at +\infty, and the existence of a second oblique asymptote
2. Show that the y-axis is an axis of symmetry for C.
3. Solve the inequation 1-e^{2x}\,\geq\,\,0.
4. Study the variations of the function f.
5. Show \Delta, \Delta' and C, after indicating the position of \Delta and C.

Cette publication est également disponible en : Français (French) العربية (Arabic)

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