# EXPONENTIAL EQUATIONS

## The mathematics lesson, exponential equations subject. Addition, subtraction, multiplication, division operations in exponential numbers. Step by step solution of exponential equations.

To solve exponential equations, the properties of the exponential numbers must be well known.

The properties used in solving the exponential equations are given below.

**1. Addition Property Of Exponential Numbers**

If the bases and exponents of the exponential numbers are the same, coefficients of the exponential numbers are summed.

**2. Equality Of Two Exponential Numbers**

If two numbers that equal coefficients and bases are equal, the exponents of these are equal.

If,

a•k^{x} = a•k^{y}

In this case,

x = y dir.

Usually, we solve the exponential equations using two properties at above.

**Example:**

2^{x} + 3•2^{x} + 5•2^{x} = 72

Find the x value in the equation above.

**Solution:**

Firstly, we must add the exponential numbers that the left side of equality.

1•2^{x} + 3•2^{x} + 5•2^{x} = (1 + 3 + 5)•2^{x}

= 9•2^{x}

Let we put this sum on the left side of the equality.

9•2^{x} = 72

Let we division by 9 the both sides of the equality.

9•2^{x} |
| |||||

9 |

2^{x} = 8

8 = 2^{3} because of,

2^{x} = 2^{3}

x = 3

**3. Multiplication Property Of The Exponential Numbers**

If bases of two exponential numbers are equal, exponents of these numbers are added. The common base remains the same. Also, the coefficients are multiplied by each other too.

If the exponent of the exponential numbers are equal but its bases are different, the bases are multiplied each other and written as a common bases. The common exponent remains the same.

k^{m} • k^{n} =k^{m+n}

x•k^{m} • y•k^{n} = x•y•k^{m + n}

**Example:**

3•5^{2x} • 2• 5^{3x} • 4•5^{x} = 600

Find the x value.

**Solution:**

3•5^{2x} • 2• 5^{3x} • 4•5^{x} = 600

3•2•4•5^{2x} • 5^{3x} • 5^{x} = 6•100

24 • 5^{(2x + 3x + x)} = 6•4•25

24•5^{6x} = 24•5^{2}

Divide the both sides of the equality by 24.

5^{6x} = 5^{2}

Because of the bases are equal, the exponents are equal too.

6x = 2

x = | 2 | |

6 |

x = | 1 | |

3 |

**Example:**

4^{2x} • 5^{2x} • 6^{2x} = 1920

Find the unknown x value.

**Solution:**

The bases are multiplied with each other and written as a common bases, the common exponent is remains the same.

4^{2x} • 5^{2x} • 6^{2x} = 1920

(4•5•6)^{2x} = 1920

120•2^{x} = 1920

Divide the both sides of the equality by 10.

12•2^{x} = 192

Now, divide the both sides by 12.

2^{x} = 16

2^{x} = 2^{4}

x = 4

**4. The Division Property Of the Exponential Numbers**

If the bases are the same, but the exponents are different, is done subtraction between exponents.

If the exponents are the same, but bases are different, it is to done division operation between the bases . The common exponent remains the same.

**Example:**

5•4^{3x} + 3•4^{3x} | = 64 | |

6•4^{2x} +2•4^{2x} |

What is unknown x value on above equation?

**Solution:**

Let we sum the terms that in the numerator and denominator.

(5 + 3)•4^{3x} | = 64 | |

(6+2)•4^{2x} |

8•4^{3x} | = 64 | |

8•4^{2x} |

The 8 that on the both sides of the equality becomes simplify.

4^{3x} | = 64 | |

4^{2x} |

Now, Let we the division operation.

4^{(3x – 2x)} = 4^{3}

4^{x} = 4^{3}

x = 3 olur.

**Example:**

Find the unknown x value on above equations.

**Solution:**

Perform addition, subtraction, multiplication, and division.

2•6^{(3x – 5x) }= 144 - 2•6^{(x – 3x)}

2•6^{-2x} = 144 – 2•6^{-2x}

4•6^{-2x} = 144

6^{-2x} = 36

6^{-2x} = 6^{2}

-2x = 2

x = -1

**Example:**

Find the value of x.

**Solution:**

On the left side of the equation, the 5 which in the numerator and the denominator are becomes simplify.

The bases of the exponential numbers that in the numerator and denominator are different, the exponents are equal. Therefore, the numerator and the denominator is done divide operation between.

Also, 3^{2x} = 9^{x }

3^{4x} = 9^{(2x – x + 1)}

3^{4x} = 3^{2(x + 1)}

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

4x = 2x + 2

2x = 2

x = 1

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29/06/2018

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