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Study Guide

It's time for our master-class before facing off with Expo and his minions. Log sends us off to an old, musty library with stack after stack of books. They've got titles like *Ye Olde Mathematical Beasts* and *Logarithmica Adeptus*. Whenever you open up one of the ancient books, dust puffs out all over your face. Don't sneeze, though; that'll just stir it up even more.

Looking through them, you find some magical properties of logarithms and exponents. Some you've already met; some are fresh and new.

**Sum of Logs: **

log* _{b} xy* = log

For all of these properties, thinking about them in their exponential form is a big help. Remember that multiplying two numbers with the same base together means we add their exponents. For example, *b ^{x}b^{y}* =

**Difference of Logs: **

This one is similar to the previous property, but in reverse gear. If two numbers with the same base are divided, their exponents are subtracted from each other.

**Exponent in Log: **

log* _{b} x^{n}* =

Whoa, how'd that exponent just decide to go outside the log? Expo just doesn't care 'bout anyone now, it seems. Zip, zoom, fly—it's time for exponent mode. Imagine the following situation:

10^{2} = 100

If we cube each side, this is what would go down:

(10^{2})^{3} = 100^{3}

The power of 3 on the right side is like *n* in the "exponent in log" rule. All you need to remember here is that when you raise an exponent to another exponent, you can multiply them:

10^{2 × 3} = 100^{3}

10^{6} = 100^{3}

Important note: we can only pull out the exponent from a log when it's attached to the entire expression inside the log. For example, we can turn log(3*x*)^{2} into 2log(3*x*). But we *can't* do anything with log(3*x*^{2}), because the exponent is only attached to the *x*, not the 3.

**Log of 1:**

log* _{b}* 1 = 0

Essentially all this property says is, "What exponent do we raise a base to so we get an answer of 1?" That only leaves one answer: 0. Absolutely any base raised to the power of 0 is 1. The base could be 1,000,000, but raising it to the 0 power would still give us 1. How mystical.

**Equal Base: **

log_{b}*b* = 1

Again, let's put on our exponential-form goggles. (Now you look like you're on a secret mission. So classy.)

The
logarithm answers the question, "What's the exponent?" If we need the
two sides to equal, the only exponent that would give a correct answer
would be 1. It'd work out just to be *b*^{1}* = b*.

Demonstrate the sum of logs by expanding log 100 and solving, then demonstrate the exponent in log property by expanding.

There are two other ways to represent log 100: log(10 × 10) or log(10^{2}).

If we expand the first way using the sum of logs rule, we get:

log(10 × 10) = log 10 + log 10 = 1 + 1 = 2

Solving log 100 the regular way also gives us 2 (remember, the common log has an invisible base of 10):

log 100 = *x*

10* ^{x}* = 100

*x* = 2

If we expand the second way using the exponent in log rule, we get:

log(10^{2}) = 2 log 10 = 2 × 1 = 2

No sweat.

Simplify the following equation using the properties you've learned so far:

3log_{4}(16*x*^{5})

First let's pull that exponent down from its pedestal:

3(5)log_{4}(16*x*) = 15log_{4}(16*x*)

Then use the sum of logs rule to pull the logs apart, making sure to hold onto the coefficient:

15(log_{4} 16 + log_{4} *x*) = 15 log_{4} 16 + 15 log_{4} *x*

Simplify the first log, because 4^{2} = 16:

15(2) + 15 log_{4} *x* = 30 + 15 log_{4} *x*

There's your answer. We can't simplify any further.

Rewrite the following logarithmic equation into a single log, simplifying where possible, using the properties you've learned.

*y = *0.5log_{5}(*x*^{2}) + log_{5}(25*x*) – 1 – log_{5} 5

Whoa boy. That's a Moby Dick-sized whale of a problem.

Let's go from left to right. First, pull the exponent out of the left log:

0.5log_{5}(*x*^{2}) = 0.5(2) log_{5} *x* = log_{5} *x*

Split the middle log into a sum:

log_{5}(25*x*) = log_{5} 25 + log_{5} *x* = 2 + log_{5} *x*

And simplify that log on the right:

log_{5} 5 = 1

Let's see what we have left:

*y* = log_{5} *x* + 2 + log_{5} *x* – 1 – 1

*y* = log_{5} *x* + log_{5} *x*

Now we can reverse the sum of logs rule and combine these into a single log:

*y* = log_{5}(*xx*)

*y* = log_{5} *x*^{2}

One more round of pulling the exponent outta the log:

*y* = 2 log_{5} *x*

Now that you've got all the skills to beat Expo at his own game, taking him down should be a cinch.

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