Exponential Equations

What are Exponential Equations?

An exponential equation is an equation in which the variable is in the exponent. The general form of an exponential equation is a^x = b, where a and b are constants and x is the variable.

Basic Properties of Exponential Equations

• An exponential function f(x) = a^x where a is a positive real number and a ≠ 1 has the following properties:
  • The domain of the function is all real numbers.
  • The range of the function is all positive real numbers.
  • The function is increasing if a > 1 and decreasing if 0 < a < 1.
  • The function has a horizontal asymptote at y = 0.

Solving Exponential Equations

To solve an exponential equation a^x = b, where a and b are constants and x is the variable, use the following steps:

1. Take the logarithm of both sides of the equation using a base that makes the computation convenient (typically base 10 or base e).
2. Apply logarithm properties to solve for the variable x.
3. Check the solution to ensure it is valid within the given domain.

Study Guide

When studying exponential equations, make sure to understand the properties of exponential functions, including their domain, range, and behavior. Practice solving exponential equations using logarithms and logarithmic properties. Also, review the rules for manipulating exponents, as they are essential for solving exponential equations.

Here are some key points to remember when studying exponential equations:

• Understand the basic properties of exponential functions.
• Practice solving exponential equations using logarithms.
• Review the rules for manipulating exponents.
• Work on real-world problems that involve exponential growth or decay.

Remember to practice solving different types of exponential equations to strengthen your understanding of the topic.