### Examples of Exponential Equations

$$ x^x = 4 \\ x^{2x} = 16 \\ x^{x+1} = 256 $$

Debug

Before you try to solve exponential equations, you must be quite comfortable using the rules and laws of exponents. An exponential equation is simply an equation in which a variable appears in the exponent.

$$ x^x = 4 \\ x^{2x} = 16 \\ x^{x+1} = 256 $$

There are many different kinds of exponential equations. First, let's focus on exponential equations that have a single term on both sides. These equations can be classified into two different types.

- Type #1) When the bases are of both terms are the same
- Type #2) When the bases are of the terms are different

Ignore the bases, and simply set the exponents equal to each other

X + 1 = 9

Step 2Solve for the variable

X = 9 − 1

X = 8

Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: *"4 and 2 are powers of what number? "* Once you have your answer, rewrite both of the bases as powers of that number)

They are both powers of 2

4 = 2²

Substitute the rewritten bases into original equation

4^{3} =2^{x}

(2^{2})^{3} =2^{x}

Simplify exponents

2^{6} = 2^{x}

Solve like an exponential equation of like bases

6 = x

Step 1

This is the second type of equation (exponents with unlike bases). So follow the Steps.

Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: *"9 and 81 are powers of what number? "* Once you have your answer, rewrite both of the bases as powers of that number)

They are both powers of 3 and of 9. You can use either base. I will use 3.

81 = 3^{4}

9 = 3^{2}

Step 2

Substitute the rewritten bases into original equation

(3^{2})^{x} =3^{4}

Step 3

Simplify exponents

3^{2}^{x} =3^{4}

Step 4

Solve like an exponential equation of like bases

2x = 4

x = 2

Step 1

Since these equations have different bases, follow the steps for unlike bases

Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: *"$$ \frac{1}{4} $$ and 32 are powers of what number? "* Once you have your answer, rewrite both of the bases as powers of that number)

They are both powers of 2

$$ 32= 2^5 \\ \frac{1}{4} = 2^{-2} \\ $$

Step 2

Substitute the rewritten bases into original equation

$$ \left(2 ^{-2} \right)^x = 2^5 $$

Step 3

Simplify exponents

$$ 2 ^{(-2 \cdot x)} = 2^5 \\ 2 ^{-2 x} = 2^5 $$

Step 4

Solve like an exponential equation of like bases

$$ -2x = 5 \\ \frac{-2x}{-2} = \frac{5}{-2} \\ x = -\frac{5}{2} $$

Step 1

Rewrite this equation so that it looks like the other ones we solved--In other words, isolate the exponential expression as follows:

$$ 4^{2x} +1 \red{-1} = 65\red{-1} \\ \bf{{4^{2x} +1 = 64}} $$

Since these equations have different bases, followthe steps for unlike bases.

Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: *"4 and 64 are powers of what number? "* Once you have your answer, rewrite both of the bases as powers of that number)

They are both powers of 2 and of 4. You could use either base to solve this. I will use base 4

$$ 4 = 4^1 \\ 64 = 4^3 $$

Step 2

Substitute the rewritten bases into original equation

$$ \bf{{4^{2x} +1 = 64}} \\ 4^{2x} = 4^3 $$

Step 3

Simplify exponents

4^{2x} = 4^{3}

Step 4

Solve like an exponential equation of like bases

$$ 2x =3 \\ x = \frac{3}{2} $$

Step 1

*" $$ \frac{1}{9}$$ and 27 are powers of what number? "* Once you have your answer, rewrite both of the bases as powers of that number)

Rewrite this equation so that it looks like the other ones we solved--In other words, isolate the exponential expression as follows: $$ \left( \frac{1}{9} \right)^x -3 \red{+3} =24\red{+3} \\ \bf{\left( \frac{1}{9} \right)^x=27 } $$

Since these equations have different bases, followthe steps for unlike bases

Rewrite the bases as powers of a common base

(Ignore the exponents and Answer the following question:They are both powers of 3.

$$ \frac{1}{9} = 3^{-2} \\ 27 = 3^3 $$

Step 2

Substitute into original equation

$$ \left( \frac{1}{9} \right)^x=27 \\ \left( \red{3^{-2}}\right)^x=\red{3^3 } $$

Step 3

Simplify exponents

$$ 3^\red{{-2 \cdot x}} = 3^3 \\ 3^\red{{-2x}} = 3^3 $$

Step 4

Solve like an exponential equation of like bases

$$ -2x = 3 \\ x = \frac{3}{-2} \\ x = -\frac{3}{2} $$

Step 1

*"1/25 and 125 are powers of what number? "* Once you have your answer, rewrite both of the bases as powers of that number)

Rewrite this equation so that it looks like the other ones we solved--In other words, isolate the exponential expression as follows:

$$ \left( \frac{1}{25} \right)^{(3x -4)} -1 \red{+1} = 124 \red{+1} \\ \left( \frac{1}{25} \right)^{(3x -4)} = 125 $$

Since these equations have different bases, followthe steps for unlike bases

Rewrite the bases as powers of a common base

(Ignore the exponents and Answer the following question:They are both powers of 5.

$$ \frac{1}{25} = 5^{-2} \\ 125 = 5^3 $$

Step 2

Substitute into original equation

$$ \left( \frac{1}{25} \right)^{(3x -4)} = 125 \\ \left( \red{5^{-2}} \right)^{(3x -4)} = \red{5^3} $$

Step 3

Simplify exponents

$$ 5^\red{{-2 \cdot (3x -4)}} = 5^3 \\ 5^\red{{(-6x + 8)}} = 5^3 $$

Step 4

Solve like an exponential equation of like bases

$$ -6x + 8 =3 \\ -6x = -5 \\ x = \frac{-5}{-6} \\ x = \frac{5}{6} $$

This page: