# Exponential Equations - Complex Equations

**EXPONENTIAL EQUATIONS: Introduction & Simple Equations.**

This discussion will focus on solving more complex problems involving exponential functions. Below is a quick review of exponential functions.

**Quick Review**

EXPONENTIAL FUNCTION

y = ab^{x}

Where a ≠ 0, b ≠ 1 and x is any real number.

The basic properties for the exponential function are:

**Property 1:**b

^{0}= 1

**Property 2:**b

^{1}= b

**Property 3:**b

^{x}= b

^{y}if and only if x = y

**One-to One Property**

**Property 4:**log

_{b}b

^{x}= x

**Inverse Property**

Let's solve some complex natural exponential equations.

Remember when solving for x, regardless of the function type, the goal is to isolate the x-variable.

^{x}) = 156

In this case divide both sides of the equation by 12. |
3 |

Since the x is an exponent of base 3, take log |
${\mathrm{log}}_{3}{3}^{x}={\text{log}}_{3}13$
Take log |

Property 4 states $lo{g}_{b}{b}^{x}=x$. Thus the left-hand side becomes x.
To get a value for log In short, take the log of base 10 of 13 and divided by the log of base 10 of 3, the original base. $lo{g}_{3}13=\frac{lo{g}_{10}13}{lo{g}_{10}3}=\frac{log13}{log3}$ |
x = log
x = log $x=\frac{\text{log}13}{\mathrm{log}3}$ Change base $x\approx 2.335$ Approximation |

**Example 1: 6(2**

^{(3x+1)}) - 8 = 52
In this case add 8 to both sides of the equation. Then divide both sides by 6. |
6(2
6(2
2 |

Since the x is an exponent of base 2, take log |
$lo{g}_{2}{2}^{3x+1}=lo{g}_{2}10$
Take log |

Property 4 states $lo{g}_{b}{b}^{x}=x$. Thus the left-hand side becomes the exponent, 3x + 1. Now isolate the x.
To get a value for log In short take the log of base 10 of 10 and divided by the log of base 10 of 2, the original base. $lo{g}_{2}10=\frac{lo{g}_{10}10}{lo{g}_{10}2}=\frac{log10}{log2}$ |
3x + 1 = log
3x = log $x=\frac{lo{g}_{2}10}{3}-\frac{1}{3}$ Divide by 3 $x=\frac{lo{g}_{2}10}{3}-\frac{1}{3}$ Exact answer $x=\frac{1}{3}\xb7\frac{\text{log}10}{\mathrm{log}2}-\frac{1}{3}$ Change base $x\approx 0.774$ Approximation |

**Example 1: 9**

^{-3-x}= 729
In this case the exponent is isolated. |
9 |

Since the x is an exponent of base 9, take log |
log |

Property 4 states $lo{g}_{b}{b}^{x}=x$. Thus the left-hand side becomes -3 - x. Now isolate the x.
To get a value for log In short take the log of base 10 of 729 and divided by the log of base 10 of 9, the original base. $lo{g}_{9}729=\frac{lo{g}_{10}729}{lo{g}_{10}9}=\frac{log729}{log9}$ |
-3 - x = log
-x = log
x = -(log
x = -(log $x=-\left(\frac{log729}{\mathrm{log}9}+3\right)$ Change base x = 6 Exact value |

Related Links:Math algebra Exponential Equations: Compound Interest Application Exponential Equations: Continuous Compound Interest Application Algebra Topics Exponential Functions |

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