exponential equation
an equation in the form y=ab∧x where a≠0, b≠1 and b>0
An exponential equation is an equation where the variable is in the exponent. The general form of an exponential equation is y = a(b^x). Here, y represents the dependent variable, x represents the independent variable, a is the initial value or starting point, and b is the growth factor or base.
Exponential equations are commonly used in finance, science, biology, and other fields where values grow or decay at a constant rate. These equations can be used to predict the future values of a variable based on its past performance or to determine the exponential growth rate of a phenomenon.
To solve an exponential equation, we often use logarithms to isolate the variable in the exponent. For example, if we have the equation 3^(x+1) = 27, we can use the logarithm base 3 to get log3(3^(x+1)) = log3(27), which simplifies to (x+1) = 3. Therefore, x = 2.
Some common applications of exponential equations include compound interest, population growth, radioactive decay, and bacterial growth. It is important to understand the properties of exponential functions, such as domain and range, asymptotes, and concavity, in order to properly analyze and interpret data.
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