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Solving Exponential Equations:
     Solving from the Definition (page 1 of 3)

Sections: Solving from the definition, Solving using logarithms, Calculators

To solve exponential equations without logarithms, you need to have equations with comparable exponential expressions on either side of the "equals" sign, so you can compare the powers and solve. For example:

• Solve 5^x = 5³.

Since the bases ("5" in each case) are the same, then the only way the two expressions could be equal is for the powers also to be the same. That is:

x = 3

This solution demonstrates how this class of equation is solved: if the bases are the same, then the powers must also be the same (in order for the two sides of the equation to be equal to each other). Since the powers are the same, you can set them equal, and solve the resulting (usually simpler) equation.

• Solve 10^1–x = 10⁴

Since the bases are the same, I can equate the powers and solve:

1 – x = 4
1 – 4 = x
–3 = x

• Solve 3^x = 9. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Since 9 = 3², this is really asking me to solve:

3^x = 3²

Since the bases are the same, I can set the powers equal:

x = 2

• Solve 3^2x–1 = 27.

In this case, I have an exponential on one side of the "equals" and a number on the other. I can solve this if I can express "27" as a power of 3. Since 27 = 3³, then I can proceed:

3^2x–1 = 27
3^2x–1 = 3³
2x – 1 = 3
2x = 4
x = 2

As you can probably tell, you will need to get good with your powers of numbers, such as the powers of 2 up through 2⁶ = 64, the powers of 3 up through 3⁵ = 243, the powers of 4 up through 4⁴ = 256, the powers of 5 up through 5⁴ = 625, the powers of 6 up through 6³ = 216, and all the squares. Don't plan to depend on your calculator; you'll want to have a certain degree of facility on the test, so familiarize yourself with the smaller powers now.

• Solve 3^x2–3x = 81.

This one works just like the previous example:

3^x2–3x = 81
3^x2–3x = 3⁴
x² – 3x = 4
x² – 3x – 4 = 0
(x – 4)(x + 1) = 0
x = –1, 4

• Solve 4^2x2+2x = 8.

This one is similar to the previous two, but not quite the same, because 8 is not a power of 4. However, both 8 and 4 are powers of 2, so convert:

4 = 2²
8 = 2³
4^2x2+2x = (2²)^2x2+2x = 2^4x2+4x

Now I can solve:

4^2x2+2x = 8
2^4x2+4x= 2³
4x² + 4x = 3
4x² + 4x – 3 = 0
(2x – 1)(2x + 3) = 0
x = ¹/₂ , ^–3/₂

• Solve 4^x+1 = ¹/₆₄.

Recall that negative exponents mean "flip the base to the other side of the fraction line". Recall also that 64 = 4³.  Then ¹/₆₄ = (4³)^–1 = 4^–3. With that, I can solve:

4^x+1 = ¹/₆₄
4^x+1 = 4^–3
x + 1 = –3
x = –4

• Solve 8 ^x–2 = sqrt[8]

Recall that square roots are one-half powers, and convert:

8 ^x–2  =  sqrt[8]
8 ^x–2  =  8 ^1/2
x – 2  =  1/2
x =  2 ¹/₂  =  ⁵/₂

By the way, the following is a common type of trick question:

• Solve 2^x = –4

Think about it: What power on the positive number "2" could possibly yield a negative number? I'll never go from positive to negative by taking powers; I can never turn a positive two into a negative anything, four or otherwise, by multiplying two by itself, regardless of the number of times I do the multiplication. Exponentiation just doesn't work that way. So the answer here is:

no solution

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