Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

An EXPONENTIAL EQUATION is an equation in which the variable appears in an exponent. A LOGARITHMIC EQUATION is an equation that involves the logarithm of an expression containing a variable. To solve exponential equations, first, see whether you can write both sides of the equation as powers of the same number. If you cannot, take the common logarithm of both sides of the equation and then apply property 7.

Using Properties of Logarithms

Copy and paste the following link into your browser to learn more about using the properties of logarithms: https://youtu.be/LwNpOWg78h4

Exponential Functions

EXPONENTIAL FUNCTIONS grow exponentially, which means they grow very, very quickly. Two squared is 4; 2³ [that is, cubed] is 8. However, by the time you get to 2⁷, you have, in four small steps from 8, already reached 128, and it only grows faster from there. Four more steps, for example, raising to 2¹¹, brings the value to 2,048. Here's what exponential functions look like: ▶︎▶︎ y = 2ⁿ The equation is y equals 2 raised to the 'nth' power. This sort of equation represents what we call "exponential growth" or "exponential decay." Other examples of exponential functions include: ▶︎▶︎ y = 3ˣ ▶︎▶︎ (x) = 4.5ˣ ▶︎▶︎ y = 2ˣ + 1 The general exponential function looks like this: y = bˣ, where the base b is any positive constant. Copy and paste the following link into your browser to learn more about solving and graphing expential functions: https://youtu.be/ls78_2UBcdY

Properties of Logarithms

Logarithmic functions and exponential functions are connected to one another in that they are INVERSES of each other. You may recall that when two functions are inverses of each other, the x and y coordinates are swapped. This leads to the most basic property involving logarithms which permit movement back and forth between logarithmic and exponential forms of an expression: ▶︎▶︎ n = logₐx is exactly the same as x = aⁿ You can use the above property to change a logarithmic expression into an exponential expression or an exponential expression into a logarithmic expression. Copy and paste the following link into your browser to learn more about properties of logarithms: https://youtu.be/L0M8rOkFOlw