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.
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Exponential equations can model real-world phenomena that exhibit exponential growth or decay, such as population growth, radioactive decay, and compound interest.
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.
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.
Graphically, the solution to an exponential equation represents the point where the exponential function intersects the horizontal line $y = b$.
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.