# Revisiting Inverse Operations

"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.

### Sample Problem

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 "exponentiate" both sides on a base of 4. Because *a*^{loga x} = *x*, the right-hand-side log will just cancel out and you'll be left with the answer.

*y* = log_{4} 8*x*

4* ^{y}* = 4

^{log4 8x}

4^{y} = 8*x*

Easy as using an electric mixer; just don't get your hair caught in it.

### Sample Problem

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.