Solve for the Power with Exponential Functions
How to Solve for the Power with Exponential Functions - Video
Example 1
Example 1:
We have two exponential functions. On the left we have 52x and on the right we have 55x+3. Luckily for us the base is the same for both. That means we can set the exponents equal to each and solve it like the equations you have seen before.
2x - 3 = 5x
-3 = 5x - 2x
-3 = 3x
x = -1
Set the exponents equal to each other.
Subtracted 2x on both sides.
Combined like terms.
Divided both sides by 3.
So, our final result is x = -1.
Now let's verify by substituting -1 back into our original equation.
52x = 55x+3
52 * (-1) = 55 * (-1) +3
5-2 = 5-5 +3
5-2 = 5-2
Substituted -1 for x.
Multiplied.
Combined like terms.
Now that we have verified that x = -1 makes the left side equal the right side we are set.
Example 2
Example 2:
We have two exponential functions. On the left we have 3x^2 and on the right we have 37x-6. Luckily for us the base is the same for both. That means we can set the exponents equal to each and solve it like the equations you have seen before.
x2 = 7x - 6
x2 - 7x + 6 = 0
(x - 1) * (x - 6) = 0
x - 1 = 0 x - 6 = 0
x = 1 x = 6
Set the exponents equal to each other.
Subtracted 7x and added 6 on both sides.
Found the linear factors.
Solved for each factor.
So, our final result is x = 1 or x = 6.
Now let's verify by substituting 1 and 6 back into our original equation.
3x^2= 37x-6
3(1)^2 = 37 *(1)-6
31 = 37-6
31 = 31
Substituted 1 for x.
Multiplied.
Combined like terms.
3x^2 = 37x-6
3(6)^2 = 37 *(6)-6
336 = 342-6
336 = 336
Substituted 1 for x.
Multiplied.
Combined like terms.
Now that we have verified that x = 1 or x = 6 make the left side equal the right side we are set.
Example 3
Example 3:
We have two exponential functions. On the left we have 93x * (1/3)5x+7 and on the right we have 27 * (3x)-4. This time we have several different bases, but do we? Each one is 3 to some power whether that power is positive or negative. So we need to change 9 to 32; (1/3) to 3-1; and 27 to 33.
Now we can simplify by using the rules of exponents.
93x * (1/3)5x+7 = 27 * (3x)-4
(32)3x * (3-1)5x+7 = (33) * (3x)-4
32*3x * 3-1*(5x+7) = 33 * 3x*(-4)
36x * 3-5x-7 = 33 * 3-4x
36x-5x-7 = 33-4x
3x-7 = 33-4x
Change 9, 1/3, and 7 to a number raised by a power.
Set up the multiplication of powers.
Multiplied.
Combined powers because same base.
Combined like terms.
Now we are back to what we did in example 1 and example 2, since we have the same base. On the left we have 3x-7 and on the right we have 33-4x. That means we can set the exponents equal to each and solve it like the equations you have seen before.
x - 7 = 3 - 4x
x = 7 + 3 - 4x
x + 4x = 10
5x = 10
x = 2
Set the exponents equal to each other.
Added 7 on both sides.
Added 4x on both sides.
Combined like terms.
Divided both sides by 5.
So, our final result is x = 2.
Now let's verify by substituting 1 and 6 back into our original equation. This time we aren't going to use the original, but the step where we have already change the numbers to 3 raised to some power.
(32)3*2 * (3-1)5*2+7 = (33) * (32)-4
(32)6 * (3-1)17 = (33) * (32)-4
312 * 3-17 = 33 * 3-8
312-17 = 33-8
3-5 = 3-5
Substituted 2 for x.
Simplified the powers on the outside.
Multiplied the powers.
Combined powers because same base.
Combined like terms.
Now that we have verified that x = 2 makes the left side equal the right side we are set.