Solving Exponential Equations

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. In other words, you have to have some base to some power equals (the same base) to (some other power), where you set the two powers equal to each other, and solve the resulting equation. i.e. If $$b^x = b^y$$ then x=y

For example:

a. Solve $5^x = 5^3$

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 entire class of equations 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 must be the same, then you can set the two powers equal to each other, and solve the resulting equation.

b. Solve $10^{1 – x} = 10^4$

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

$$1- x = 4 $$

$$1 – 4 = x $$

$$-3 = x$$

Sometimes you’ll first need to convert one side or the other (or both) to some other base before you can set the powers equal to each other. For example:

c. Solve $3^x = 9$

Since $9 = 3^2$, this is really asking me to solve:

$$3^x = 3^2$$

By converting the 9 to a $3^2$, I’ve converted the right-hand side of the equation to having the same base as the left-hand side. Since the bases are now the same, I can set the two powers equal to each other:

$x = 2$

d. 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 the equation if I can express the 27 as a power of 3. Since $27 = 3^3$, then I can convert and proceed with the solution:

$$3^{2x – 1} = 27 $$

$$3^{2x – 1} = 3^3$$

$$2x – 1 = 3 $$

$$2x = 4 $$

$$x = 2$$

**Exercise**

Solve the following exponential equations:

1. $3x^2 – 3x = 81$

2. $4^{2x^2+2x} = 8$

3. $4^{x+1} = \frac{1}{64}$

4. $8^{x – 2} = \sqrt{8}$

5. $2^x = {-4}$

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