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You get a message dropped through a crack in your window in the middle of the night. It reads, "I am ready to lend a hand in the fight against the evil Expo. Meet me in the park at noon tomorrow." When you get there the next day, it's none other than Ms. Log Arithm! She greets you warmly and jumps right to business.

### 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

= 4^{y}^{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.

### Change of Base

Sometimes you'll want to solve a logarithm that isn't base-10 or a natural log. "That's crazy! Madness!" you might scream. We can empathize. If you want to get Log's help in the fight with Expo, though, there's going to be some long, strenuous training sessions ahead. Cue training montage music.Most calculators only have buttons for common logs (log) and natural logs (ln), so calculating a logarithm with a different base can be a real pain the neck. Thankfully, the tool you need to solve these bizarro-logs is pretty easy to use.

Like ye olde alchemists of medieval times (not Medieval Times the restaurant, the actual time period), you can actually change the base of any logarithm. Not quite as impressive as changing lead into gold, but we do what we can.

Let's say we've got this nasty-looking lump of log:

log

_{7}100After adding a few powdered pearls, some fire salts, maybe a little old cheese, and some heat, we can use this formula:

PLEH! The pearl powder got all over us. We guess we'd better turn the heat down. We can see that this formula allows us to change a log of any base into a fraction of two logs that have a new base of the same value. This new base can be anything we want, as long as it's a proper logarithm base (that is, it's not zero or negative). We'll go easy on ourselves and use base-10.

Let's get on to turning that log to gold:

Maybe 2.367 isn't exactly gold, but it'll have to do for now. At the very least, you're one step closer to being able to handle logarithmic functions.

### Limits of Logarithmic Functions

Just like exponential functions,**logarithmic functions**have their own limits. Remember what exponential functions can't do: they can't output a negative number for*f*(*x*). The function we took a gander at when thinking about exponential functions was*f*(*x*) = 4.^{x}Let's hold up the mirror by taking the base-4 logarithm to get the inverse function:

*f*(*x*) = log_{4}*x*.If we tried to make

*x*negative or zero in this log function, there is no*y*-value in the known universe that would let us do itâ€”so the log function is undefined at*x*-values of zero or less. In other words, its domain is*x*> 0.Here's what the graph of

*f*(*x*) = log_{4}*x*looks like:Because the

*output*of an exponential function can never be zero or negative, the inverse (log) function can never have a negative*input*of zero.### Sample Problem

When will

*f*(*x*) = log_{5}*x*be greater than*g*(*x*) = log_{20}*x*? Ignore negative*x*-values.The output of these logs is the

*exponent*needed above 5 or 20 to equal*x.*When

*x*is greater than 1, log_{5}*x*will be larger because a*larger exponent*is needed for 5 than for 20 to equal whatever*x*is in this domain. The 5 is a little shrimp that needs a boost, basically. For example:For

*x*= 2, we get 5^{0.43}or 20^{0.23}.

For*x*= 5, we get 5^{1}or 20^{0.54}.However, when

*x*is less than 1, log_{20}*x*will be greater than log_{5}*x*because a negative exponent closer to zero is needed to pull that 20 down to size versus the 5. That 20 has a lot of muscle, and needs a big push by the exponent to get down to size.For , we get 5

^{-0.43}or 20^{-0.23}.For , we get 5

^{-0.86}or 20^{-0.46}.

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