Logarithms: corrected high school math exercises 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:

definition of the logarithm;
functional equations;
algebraic formulas on logarithms;
limits and logarithm functions.

Exercise #1:

Solve the following equations:

$a)e^x=1\\b)e^x=4\\c)e^{2x}=2\\d)4e^{-x}-3=12\\e)e^{2x-1}=2\\f)e^{-x}=-5$

Exercise #2:

Solve the following equations:

$a)lnx=3\\b)ln(2x)=0\\c)2lnx-1=6\\d)ln(2x-1)=3\\e)ln\,(\,\frac{1}{x-1}\,)=1\,\\f)ln(x^2)=4$

Exercise #3:

Simplify the writing of the following numbers:

$a=ln3+ln\frac{1}{3}\\b=ln\frac{1}{16}\\c=\frac{1}{2}ln\sqrt{2}$

Exercise #4:

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

$a)ln(1-x)+ln(1+x)=2(ln2-ln5)\\b)lnx+ln(x-3)=ln4\\c)ln(x(x-3))=ln4$

Exercise #5:
Let f be the function defined on $%5D0;+\infty%5B$ by: $f(x)=2x-1+ln\,(\,\frac{x}{x+1}\,\,)$ .
We note 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 is an asymptote to in $+\infty$ .
Study the position of 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 .

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 $pH\,=\,-\,log%5BH_3O^{+}%5D$ where is the concentration of ions , expressed in mol/L.

1. A solution has a concentration of ions equal to $5\times \,10^{-9}$ .
What is its pH? What can we say about a solution whose ion concentration is ?
equal to 0.1 ?
2. What is the ion concentration of a neutral solution?
3. If you increase the concentration of 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 :
$f(x)=x+2-ln(1+e^{2x})$ .
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 :
$f(x)=2-x-ln(1+e^{-2x})$
d. Determine the limit of f at $+\infty$ , and the existence of a second oblique asymptote
$\Delta'$ .
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.

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Logarithm: senior math exercises corrected in PDF.

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