- Topics At a Glance
- Linking Exponents and Logarithms
- Inverse Functions
- Rules for Inverse Functions
- The Base
- The Natural Log
- Exponential Functions
- Linear and Exponential Growth
- Exponential Growth and Decay
- Solving Exponential Equations
- Limits of Exponential Functions
**Logarithmic Functions****Revisiting Inverse Operations**- Change of Base
- Limits of Logarithmic Functions
- Properties of Exponents and Logarithms
- In the Real World

"Remember, when my base is right I can cancel out Expo's menace," Log tells you. "Here's the second important skill you'll need, 'Summon the Inverse Operation!'"

log_{a}*a*^{x} = *x**a*^{loga x} = *x*

These special properties exist because logs and exponents are inverse operations of one another. Remember? It was from before Expo turned out to be an evil wizard lord.

What? Expo erased your memory with his magical powers? Figures.

Which of the following functions is equivalent to *y* = log_{4} 8*x*

(A) 4^{x} = 8*y*

(B) *x*^{4} = 8*y*

(C) *y*^{8} = 4*x*

(D) 4^{y} = 8*x*

Remember that the number subscript attached to the log is called the base. The left hand side of the log equation, *y, *is the exponent needed to equal 8*x*. So, the answer is **(D)**! Another easier way to solve this is just to take both sides to the 4^{th} power. Because *a*^{loga x} = *x*, the right hand side log will just cancel out and you'll be left with the answer. Easy as using an electric mixer, just don't get your hair caught in it.

Solve the following equation for *y. *Remember that exponentiation and taking logs are operations that you can apply to both sides of an equation:

Because this log has a base of 5, we have to exponentiate (take each side to some power) using a matching base.

^{}

Notice that this equation is an exponential function.