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Fonction exponentielle
Savoir résoudre des équations avec les exponentielles - Exercice 1
10 min
15
Résoudre les équations suivantes sur
R
\mathbb{R}
R
.
Question 1
e
2
x
=
e
4
e^{2x} =e^{4}
e
2
x
=
e
4
Correction
e
A
=
e
B
⇔
A
=
B
e^{A} =e^{B} \Leftrightarrow A=B
e
A
=
e
B
⇔
A
=
B
e
2
x
=
e
4
e^{2x} =e^{4}
e
2
x
=
e
4
équivaut successivement à :
2
x
=
4
2x=4
2
x
=
4
x
=
4
2
x=\frac{4}{2}
x
=
2
4
x
=
2
x=2
x
=
2
Donc
S
=
{
2
}
S=\left\{2 \right\}
S
=
{
2
}
Question 2
e
3
x
+
12
=
1
e^{3x+12} =1
e
3
x
+
12
=
1
Correction
e
0
=
1
e^{0} =1
e
0
=
1
e
A
=
e
B
⇔
A
=
B
e^{A} =e^{B} \Leftrightarrow A=B
e
A
=
e
B
⇔
A
=
B
e
3
x
+
12
=
1
e^{3x+12} =1
e
3
x
+
12
=
1
équivaut successivement à :
e
3
x
+
12
=
e
0
e^{3x+12} =e^{0}
e
3
x
+
12
=
e
0
3
x
+
12
=
0
3x+12=0
3
x
+
12
=
0
3
x
=
−
12
3x=-12
3
x
=
−
12
x
=
−
12
3
x=\frac{-12}{3}
x
=
3
−
12
x
=
−
4
x=-4
x
=
−
4
Donc
S
=
{
−
4
}
S=\left\{-4\right\}
S
=
{
−
4
}
Question 3
e
5
x
=
−
4
e^{5x} =-4
e
5
x
=
−
4
Correction
e
5
x
=
−
4
e^{5x} =-4
e
5
x
=
−
4
Equation impossible à résoudre car une exponentielle est
strictement positive.
S
=
∅
S=\emptyset
S
=
∅
Question 4
e
−
8
x
+
2
=
e
e^{-8x+2} =e
e
−
8
x
+
2
=
e
Correction
e
A
=
e
B
⇔
A
=
B
e^{A} =e^{B} \Leftrightarrow A=B
e
A
=
e
B
⇔
A
=
B
e
−
8
x
+
2
=
e
e^{-8x+2} =e
e
−
8
x
+
2
=
e
équivaut successivement à :
e
−
8
x
+
2
=
e
1
e^{-8x+2} =e^{1}
e
−
8
x
+
2
=
e
1
−
8
x
+
2
=
1
-8x+2=1
−
8
x
+
2
=
1
−
8
x
=
1
−
2
-8x=1-2
−
8
x
=
1
−
2
−
8
x
=
−
1
-8x=-1
−
8
x
=
−
1
x
=
−
1
−
8
x=\frac{-1}{-8}
x
=
−
8
−
1
x
=
1
8
x=\frac{1}{8}
x
=
8
1
Donc
S
=
{
1
8
}
S=\left\{\frac{1}{8}\right\}
S
=
{
8
1
}
Question 5
e
2
x
×
e
5
x
=
e
14
e^{2x} \times e^{5x} =e^{14}
e
2
x
×
e
5
x
=
e
14
Correction
e
A
×
e
B
=
e
A
+
B
e^{A}\times e^{B}=e^{A+B}
e
A
×
e
B
=
e
A
+
B
e
A
=
e
B
⇔
A
=
B
e^{A} =e^{B} \Leftrightarrow A=B
e
A
=
e
B
⇔
A
=
B
e
2
x
×
e
5
x
=
e
14
e^{2x} \times e^{5x} =e^{14}
e
2
x
×
e
5
x
=
e
14
e
2
x
+
5
x
=
e
14
e^{2x+5x} =e^{14}
e
2
x
+
5
x
=
e
14
e
7
x
=
e
14
e^{7x} =e^{14}
e
7
x
=
e
14
7
x
=
14
7x=14
7
x
=
14
x
=
14
7
x=\frac{14}{7}
x
=
7
14
x
=
2
x=2
x
=
2
Donc
S
=
{
2
}
S=\left\{2\right\}
S
=
{
2
}
Question 6
(
e
3
x
+
1
)
2
=
e
x
−
1
\left(e^{3x+1} \right)^{2} =e^{x-1}
(
e
3
x
+
1
)
2
=
e
x
−
1
Correction
(
e
A
)
B
=
e
A
×
B
\left(e^{A} \right)^{B} =e^{A\times B}
(
e
A
)
B
=
e
A
×
B
e
A
=
e
B
⇔
A
=
B
e^{A} =e^{B} \Leftrightarrow A=B
e
A
=
e
B
⇔
A
=
B
(
e
3
x
+
1
)
2
=
e
x
−
1
\left(e^{3x+1} \right)^{2} =e^{x-1}
(
e
3
x
+
1
)
2
=
e
x
−
1
équivaut successivement à :
e
(
3
x
+
1
)
×
2
=
e
x
−
1
e^{\left(3x+1\right)\times 2} =e^{x-1}
e
(
3
x
+
1
)
×
2
=
e
x
−
1
e
6
x
+
2
=
e
x
−
1
e^{6x+2} =e^{x-1}
e
6
x
+
2
=
e
x
−
1
6
x
+
2
=
x
−
1
6x+2=x-1
6
x
+
2
=
x
−
1
6
x
=
x
−
1
−
2
6x=x-1-2
6
x
=
x
−
1
−
2
6
x
=
x
−
3
6x=x-3
6
x
=
x
−
3
6
x
−
x
=
−
3
6x-x=-3
6
x
−
x
=
−
3
5
x
=
−
3
5x=-3
5
x
=
−
3
x
=
−
3
5
x=-\frac{3}{5}
x
=
−
5
3
Donc
S
=
{
−
3
5
}
S=\left\{-\frac{3}{5}\right\}
S
=
{
−
5
3
}
Question 7
e
2
x
e
5
x
−
2
=
e
−
4
x
+
8
\frac{e^{2x} }{e^{5x-2} } =e^{-4x+8}
e
5
x
−
2
e
2
x
=
e
−
4
x
+
8
Correction
e
A
e
B
=
e
A
−
B
\frac{e^{A}}{ e^{B}}=e^{A-B}
e
B
e
A
=
e
A
−
B
e
A
=
e
B
⇔
A
=
B
e^{A} =e^{B} \Leftrightarrow A=B
e
A
=
e
B
⇔
A
=
B
e
2
x
e
5
x
−
2
=
e
−
4
x
+
8
\frac{e^{2x} }{e^{5x-2} } =e^{-4x+8}
e
5
x
−
2
e
2
x
=
e
−
4
x
+
8
équivaut successivement à :
e
2
x
−
(
5
x
−
2
)
=
e
−
4
x
+
8
e^{2x-\left(5x-2\right)} =e^{-4x+8}
e
2
x
−
(
5
x
−
2
)
=
e
−
4
x
+
8
e
2
x
−
5
x
+
2
=
e
−
4
x
+
8
e^{2x-5x+2} =e^{-4x+8}
e
2
x
−
5
x
+
2
=
e
−
4
x
+
8
e
−
3
x
+
2
=
e
−
4
x
+
8
e^{-3x+2} =e^{-4x+8}
e
−
3
x
+
2
=
e
−
4
x
+
8
−
3
x
+
2
=
−
4
x
+
8
-3x+2=-4x+8
−
3
x
+
2
=
−
4
x
+
8
−
3
x
=
−
4
x
+
8
−
2
-3x=-4x+8-2
−
3
x
=
−
4
x
+
8
−
2
−
3
x
=
−
4
x
+
6
-3x=-4x+6
−
3
x
=
−
4
x
+
6
−
3
x
+
4
x
=
6
-3x+4x=6
−
3
x
+
4
x
=
6
x
=
6
x=6
x
=
6
Donc
S
=
{
6
}
S=\left\{6\right\}
S
=
{
6
}