EEE - Solve Exponential Equations (Lesson)

Solve Exponential Equations

We can apply the properties of exponents you've learned to solve equations involving exponents. You just need to know one more property:

Property	In words:	Example:
If b > 0 and b does not equal 1, then b^x = b^y, if and only if x = y.	Two powers with the same positive base other than 1 are equal if and only if the exponents are equal.	If 2^x = 2⁹, then x = 9.

Example 1:

3^x = 81

3^x = (3)⁴	We know that 3⁴ = 81 so we can replace 81 in the equation.
x = 4	Using our property above, we know that x = 4.

Example 2:

$LaTeX: \frac{5}{2}(2)^x=80$ $\frac{5}{2}(2)^x=80$

$LaTeX: \frac{2}{5}\cdot\frac{5}{2}\left(2\right)^x=80\cdot\frac{2}{5}$ $\frac{2}{5}\cdot\frac{5}{2}\left(2\right)^x=80\cdot\frac{2}{5}$	Multiply by (2/5) on either side to isolate the base and exponent.
2^x = 32	Simplify
2^x = 2⁵	We know that 2⁵ = 32 so we can replace 32 in the equation.
x = 5	Using our property above, we know that x = 5.

Watch this video to try a few more:

Try these problems to see if you've got it. Solve for x.

$LaTeX: \frac{3}{4}\left(7\right)^x=\frac{147}{4}$ $\frac{3}{4}\left(7\right)^x=\frac{147}{4}$
$LaTeX: 5\left(\frac{1}{2}\right)^x=\frac{5}{8}$ $5\left(\frac{1}{2}\right)^x=\frac{5}{8}$
3(2)^x = 384

Let's try a more challenging problem:

Example 3:

3^x+2 = 81^2x

3^x+2 = (3⁴)^2x	We know that 3⁴ = 81 so we can replace 81 in the equation.
3^x+2 = (3^8x)	Using our properties of exponents, we can multiply 4(2x) = 8x.
x +2 = 8x	We set the exponents equal to one another: b^x = b^y if and only if x = y.
x + 2 = 8x 2 = 7x (2/7) = x	Solve for x.

Watch this video to try a few more:

Try these problems to see if you've got it. Solve for x.

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