rozwiązywanie równan wykladniczych prosze o pomoc
z.716
a) 4^x * 2^x = 64
(2 ^2)^x * 2^x = 2^6
2^(2x) *2^x = 2^6
2^(2x + x) = 2^6
3x = 6
x = 2
=====
b)
0,25 ^x * 8^(x -1) = 2
(1/4)^x * ( 2^3)^(x -1) = 2
[(1/2)^2]^x * 2^(3x) = 2
(1/2)^(2x) * 2^(3x) = 2
[ 2 ^(-1)]^(2x) * 2^(3x) = 2
2^(-2x) * 2^(3x) = 2
2^(-2x +3x) = 2
2^x = 2
x = 1
======
c)
10^x *100^x *1000^x = 0,1^(1 -x)
10^x * (10^2)^x * (10^3)^x = [(10^(-1)]^(1 -x)
10^x * 10^(2x) * 10^(3x) = 10 ^(x -1)
10^(x +2x +3x) = 10^(x -1)
10^(6x) = 10^(x -1)
6x = x - 1
5x = - 1
x = -1/5
=============
d)
[ p(2)]^x * [ p(8)]^x = 32
[ p(2)]^x * [ p(2)^3]^x = 32
[ p(2)]^x * [ p(2)]^3x = 32
[ p(2)]^(4x) = 32
[ (p(2)^2]^(2x) = 2^5
2^(2x) = 2^5
2x = 5
x = 2,5
==========
e)
[ 10 p(1000)]^x * 0,1^x = [ p5st(0,1)]^(3x + 2)
[ 10 * 10 p(10)]^x * 0,1 ^x = [ 0,1^(1/5)]^(3x + 2)
[100 * 10^(1/2)]^x * [ 10^(-1)]^x = [ (10^(-1)) ^(1/5)]^(3x + 2)
[ 10^2 * 10^(1/2)]^x * 10^(-x) = [ 10^(-1/5)]^(3x + 2)
[10^2,5]^x * 10^(-x) = 10 ^(-0,6x - 0,4)
10^2,5x * 10^(-x) = 10 ^( -0,6x - 0,4)
10^(1,5x) = 10^( -0,6 x - 0,4)
1,5x = -0,6x - 0,4
2,1 x = - 0,4
21x = - 4
x = - 4/21
===========
f)
7^x *(1/49)^(2x) = 7* [ p3st(7)/7]^x
7^x *( 7^(-2)]^(2x) = 7*[ 7^(1/3) / 7]^x
7^x * 7^(-4x) = 7*[ 7^(-2/3)]^x
7^(-3x) = 7*7^[( -2/3) x]
7^(-3x) = 7^[ (-2/3) x + 1]
-3x = (-2/3)x + 1
( -7/3) x = 1
x = - 3/7
z.717
a)
2^x + 2^(x +2) = 20
2^x + 2^2 * 2^x = 20
2^x +4 *2^x = 20
5* 2^x = 20 / : 5
2^x = 4
========
3^x - 3^(x-1) = 54
3^x - (1/3)*3^x = 54
(2/3)* 3^x = 54 / * 3/2
3^x = 81
3^x = 3^4
x = 4
================
10^(2x + 1) - 10^(2x -1) =990 p(10)
10^10^(2x) - (1/10)*10^(2x) = 990 p(10)
9,9 *10^(2x) = 990 p(10) / : 9,9
10^(2x) = 100 p(10)
10^(2x ) = 10^2 * 10^(1/2)
10^(2x) = 10^2,5
2x =2,5
x = 1,25 = 5/4
=================
9^(x/2) + 3^(x +2) = p( 2 700)
(3^2)^(x/2) + 3^2 * 3^x = p( 9*3*100)
3^x +9* 3^x = 3*10*p(3)
10* 3^x = 10* 3^1 * 3^(1/2) / : 10
3^x = 3^(1,5)
x =1,5
===================
5^(2x) + 25^(x +1) = 1,04
(5^2)^x + 25 *25^x = 1 1/25
25^x + 25 *25^x = 26/25
26* 25^x = 26/25 / : 26
25^x - 1/ 25
25^x = 25^(-1)
x = - 1
====================
1/( 7^x) + 7^(2 - x) - 350
7^(-x) + 7^2 * 7^(-x) = 350
7^(-x) + 49 *7^(-x) = 350
50 * 7^(-x) = 350 / : 50
7^(-x) = 7
-x = 1
===============
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z.716
a) 4^x * 2^x = 64
(2 ^2)^x * 2^x = 2^6
2^(2x) *2^x = 2^6
2^(2x + x) = 2^6
3x = 6
x = 2
=====
b)
0,25 ^x * 8^(x -1) = 2
(1/4)^x * ( 2^3)^(x -1) = 2
[(1/2)^2]^x * 2^(3x) = 2
(1/2)^(2x) * 2^(3x) = 2
[ 2 ^(-1)]^(2x) * 2^(3x) = 2
2^(-2x) * 2^(3x) = 2
2^(-2x +3x) = 2
2^x = 2
x = 1
======
c)
10^x *100^x *1000^x = 0,1^(1 -x)
10^x * (10^2)^x * (10^3)^x = [(10^(-1)]^(1 -x)
10^x * 10^(2x) * 10^(3x) = 10 ^(x -1)
10^(x +2x +3x) = 10^(x -1)
10^(6x) = 10^(x -1)
6x = x - 1
5x = - 1
x = -1/5
=============
d)
[ p(2)]^x * [ p(8)]^x = 32
[ p(2)]^x * [ p(2)^3]^x = 32
[ p(2)]^x * [ p(2)]^3x = 32
[ p(2)]^(4x) = 32
[ (p(2)^2]^(2x) = 2^5
2^(2x) = 2^5
2x = 5
x = 2,5
==========
e)
[ 10 p(1000)]^x * 0,1^x = [ p5st(0,1)]^(3x + 2)
[ 10 * 10 p(10)]^x * 0,1 ^x = [ 0,1^(1/5)]^(3x + 2)
[100 * 10^(1/2)]^x * [ 10^(-1)]^x = [ (10^(-1)) ^(1/5)]^(3x + 2)
[ 10^2 * 10^(1/2)]^x * 10^(-x) = [ 10^(-1/5)]^(3x + 2)
[10^2,5]^x * 10^(-x) = 10 ^(-0,6x - 0,4)
10^2,5x * 10^(-x) = 10 ^( -0,6x - 0,4)
10^(1,5x) = 10^( -0,6 x - 0,4)
1,5x = -0,6x - 0,4
2,1 x = - 0,4
21x = - 4
x = - 4/21
===========
f)
7^x *(1/49)^(2x) = 7* [ p3st(7)/7]^x
7^x *( 7^(-2)]^(2x) = 7*[ 7^(1/3) / 7]^x
7^x * 7^(-4x) = 7*[ 7^(-2/3)]^x
7^(-3x) = 7*7^[( -2/3) x]
7^(-3x) = 7^[ (-2/3) x + 1]
-3x = (-2/3)x + 1
( -7/3) x = 1
x = - 3/7
===========
z.717
a)
2^x + 2^(x +2) = 20
2^x + 2^2 * 2^x = 20
2^x +4 *2^x = 20
5* 2^x = 20 / : 5
2^x = 4
x = 2
========
b)
3^x - 3^(x-1) = 54
3^x - (1/3)*3^x = 54
(2/3)* 3^x = 54 / * 3/2
3^x = 81
3^x = 3^4
x = 4
================
c)
10^(2x + 1) - 10^(2x -1) =990 p(10)
10^10^(2x) - (1/10)*10^(2x) = 990 p(10)
9,9 *10^(2x) = 990 p(10) / : 9,9
10^(2x) = 100 p(10)
10^(2x ) = 10^2 * 10^(1/2)
10^(2x) = 10^2,5
2x =2,5
x = 1,25 = 5/4
=================
d)
9^(x/2) + 3^(x +2) = p( 2 700)
(3^2)^(x/2) + 3^2 * 3^x = p( 9*3*100)
3^x +9* 3^x = 3*10*p(3)
10* 3^x = 10* 3^1 * 3^(1/2) / : 10
3^x = 3^(1,5)
x =1,5
===================
e)
5^(2x) + 25^(x +1) = 1,04
(5^2)^x + 25 *25^x = 1 1/25
25^x + 25 *25^x = 26/25
26* 25^x = 26/25 / : 26
25^x - 1/ 25
25^x = 25^(-1)
x = - 1
====================
f)
1/( 7^x) + 7^(2 - x) - 350
7^(-x) + 7^2 * 7^(-x) = 350
7^(-x) + 49 *7^(-x) = 350
50 * 7^(-x) = 350 / : 50
7^(-x) = 7
-x = 1
x = - 1
===============