Exponential functions are equations in which the variable appears in the exponent, while logarithmic functions are equations in which the variable appears in the base. In other words, exponential functions represent growth or decay, while logarithmic functions represent the inverse of exponential growth or decay.
To solve an exponential equation, you must use logarithms. If the equation is in the form of b^x = y, take the logarithm of both sides with base b to cancel out the exponent and solve for x. To solve a logarithmic equation, you must use exponentials. If the equation is in the form of log base b of x = y, rewrite it as b^y = x and solve for x.
Exponential functions are used to model growth or decay in many fields such as population growth, compound interest, and radioactive decay. Logarithmic functions are used to represent data that grows exponentially, such as the Richter scale for earthquakes, pH scale for acidity, and decibel scale for sound intensity.
Yes, both exponential and logarithmic functions can be graphed. Exponential functions have a characteristic shape of a curve that either increases or decreases rapidly, depending on the base. Logarithmic functions have a characteristic shape of a curve that starts off with a steep slope and then levels off as the input increases.
Yes, there are a few rules and properties that can help simplify and manipulate exponential and logarithmic equations. These include the product rule, quotient rule, power rule, and change of base formula for logarithms. It is important to follow these rules when solving equations involving exponential and logarithmic functions.