Exponential Equation

Definition

An exponential equation is a mathematical equation in which the variable appears as the exponent of a constant. These equations involve the exponential function, where the value of a variable is raised to a power, and the solution involves finding the value of the variable that satisfies the equation.

5 Must Know Facts For Your Next Test

1. Exponential equations can model real-world phenomena that exhibit exponential growth or decay, such as population growth, radioactive decay, and compound interest.
2. The general form of an exponential equation is $a^x = b$, where $a$ is the base, $x$ is the variable, and $b$ is the constant.
3. Solving exponential equations often involves using the properties of logarithms to convert the equation to a linear form, making it easier to isolate the variable.
4. Graphically, the solution to an exponential equation represents the point where the exponential function intersects the horizontal line $y = b$.
5. Exponential equations can have one, two, or no solutions, depending on the values of the base, variable, and constant.

Review Questions

• Explain how the properties of logarithms can be used to solve exponential equations.
  • To solve an exponential equation of the form $a^x = b$, we can apply the properties of logarithms to convert the equation to a linear form. By taking the logarithm of both sides, we get $\log_a(a^x) = \log_a(b)$, which simplifies to $x = \log_a(b)$. This allows us to isolate the variable $x$ and find its value that satisfies the original exponential equation.
• Describe the relationship between exponential functions and exponential equations, and how they are used to model real-world phenomena.
  • Exponential functions and exponential equations are closely related. Exponential functions, which take the form $f(x) = a^x$, where $a$ is the base, are used to model situations that exhibit exponential growth or decay. Exponential equations, which take the form $a^x = b$, are used to find the value of the variable $x$ that satisfies the equation. These equations are often used to model real-world phenomena, such as population growth, radioactive decay, and compound interest, where the rate of change is proportional to the current value.
• Analyze the factors that determine the number of solutions to an exponential equation and how the graph of the equation can be used to visualize the solutions.
  • The number of solutions to an exponential equation $a^x = b$ depends on the values of the base $a$ and the constant $b$. If $a > 1$ and $b > 0$, the equation will have one solution. If $a < 1$ and $b > 0$, the equation will have no solutions. If $a > 1$ and $b < 0$, the equation will have two solutions. Graphically, the solutions to the exponential equation represent the points where the exponential function $f(x) = a^x$ intersects the horizontal line $y = b$. The number of intersections determines the number of solutions, which can be visualized by plotting the graph of the exponential function and the horizontal line.

Related terms

Exponential Function: A function that describes an exponential relationship, where the independent variable appears as the exponent of a constant base value.

Logarithm: The inverse operation of exponentiation, used to solve exponential equations by converting them to linear equations.

Exponential Growth: A pattern of growth where the rate of change is proportional to the current value, resulting in a curve that increases at an accelerating rate.