To solve exponential equations, we have to follow the steps given below.
Step 1 :
If we see more than one terms in any of the sides, using properties of exponents we have to convert them as one term.
Step 2 :
Try to write the bases on both sides of equal signs as same base.
Step 3 :
Example 1 :
Solve 23x-4 = 8x-1
Solution :
43x-4 = 8x-1
Trying to write the bases on both sides of the equal sign as same base.
43x-4 = 23(x-1)
22(3x-4) = 23x - 3
Now we get the same bases on both sides. So, equating the powers. We get,
6x - 8 = 3x - 3
6x - 3x = -3 + 8
3x = 5
x = 5/3
So, the value of x is 5/3.
Example 2 :
Solve 3x-4 = 9√3
Solution :
3x-4 = 9√3
3x-4 = 32√3
3x-4 = 32+(1/2)
3x-4 = 3(5/2)
x - 4 = 5/2
x = 5/2 + 4
x = 13/2
So, the value of x is 13/2.
Example 3 :
Solve 9x+2 = 27x
Solution :
9x+2 = 27x
32(x+2) = 33x
32x + 4 = 33x
Equating the powers, we get
2x + 4 = 3x
2x - 3x = -4
-x = -4
x = 4
So, the value of x is 4.
Example 4 :
Solve 216x = 6x+10
Solution :
216x = 6x+10
63x = 6x+10
3x = x + 10
3x - x = 10
2x = 10
x = 10/2
x = 5
So, the value of x is 5.
Example 5 :
Solve 36-3x+3 = (1/216)x+10
Solution :
36-3x+3 = (1/216)x+10
62(-3x+3) = (1/6)3(x+10)
62(-3x+3) = 6-3(x+10)
Equating the powers, we get
2(-3x + 3) = -3(x + 10)
-6x + 18 = -3x - 30
-6x + 3x = -30 - 18
-3x = -48
x = 48/3
x = 16
So, the value of x is 16.
Example 6 :
Solve 92x = 27x+4
Solution :
92x = 27x+4
Writing 9 and 27 in exponential form with the same base.
(32)2x = (33)x+4
34x = 33x+12
4x = 3x + 12
4x - 3x = 12
x = 12
So, the value of x is 12.
Example 7 :
Solve 81x = 243x+2
Solution :
81x = 243x+2
(34)x = (35)x+2
34x = 35(x+2)
4x = 5x + 10
4x - 5x = 10
-x = 10
x = -10
So, the value of x is -10
Example 8 :
Solve for n
Solution :
17n + 8 = 4
17n = 4 - 8
17n = -4
n = -4/17
So, the value of n is -4/17.
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