Working with Exponents and Logarithms
What is an Exponent?
The exponent of a number says how many times In this example: 2^{3} = 2 × 2 × 2 = 8 (2 is used 3 times in a multiplication to get 8) 
What is a Logarithm?
A Logarithm goes the other way.
It asks the question "what exponent produced this?":
And answers it like this:
In that example:
 The Exponent takes 2 and 3 and gives 8 (2, used 3 times in a multiplication, makes 8)
 The Logarithm takes 2 and 8 and gives 3 (2 makes 8 when used 3 times in a multiplication)
A Logarithm says how many of one number to multiply to get another number
So a logarithm actually gives us the exponent as its answer:
(Also see how Exponents, Roots and Logarithms are related.)Working Together
Exponents and Logarithms work well together because they "undo" each other (so long as the base "a" is the same):
They are "Inverse Functions"
Doing one, then the other, gets us back to where we started:
It is too bad they are written so differently ... it makes things look strange. So it may help to think of a^{x} as "up" and log_{a}(x) as "down":
Anyway, the important thing is that:
The Logarithmic Function is "undone" by the Exponential Function.
(and vice versa)
Like in this example:
Example, what is x in log_{3}(x) = 5
We want to "undo" the log_{3} so we can get "x ="
And also:
Example: Calculate y in y = log_{4}(\frac{1}{4})
Now a simple trick: \frac{1}{4} = 4^{1}
Properties of Logarithms
One of the powerful things about Logarithms is that they can turn multiply into add.
log_{a}( m × n ) = log_{a}m + log_{a}n
"the log of multiplication is the sum of the logs"
Why is that true? See Footnote.
Using that property and the Laws of Exponents we get these useful properties:
log_{a}(m × n) = log_{a}m + log_{a}n  the log of multiplication is the sum of the logs 
log_{a}(m/n) = log_{a}m − log_{a}n  the log of division is the difference of the logs 
log_{a}(1/n) = −log_{a}n  this just follows on from the previous "division" rule, because log_{a}(1) = 0 
log_{a}(m^{r}) = r ( log_{a}m )  the log of m with an exponent r is r times the log of m 
Remember: the base "a" is always the same!
History: Logarithms were very useful before calculators were invented ... for example, instead of multiplying two large numbers, by using logarithms we can turn it into addition (much easier!)
And there were books full of Logarithm tables to help.
Let us have some fun using the properties:
Example: Simplify log_{a}( (x^{2}+1)^{4}√x )
That is as far as we can simplify it ... we can't do anything with log_{a}(x^{2}+1)
Answer: 4 log_{a}(x^{2}+1) + ½ log_{a}(x)
Note: there is no rule for handling log_{a}(m+n) or log_{a}(m−n)
We can also apply the logarithm rules "backwards" to combine logarithms:
Example: Turn this into one logarithm: log_{a}(5) + log_{a}(x) − log_{a}(2)
Answer: log_{a}(5x/2)
The Natural Logarithm and Natural Exponential Functions
When the base is Euler's Number e = 2.718281828459... we get:
 The Natural Logarithm log_{e}(x) which is more commonly written ln(x)
 The Natural Exponential Function e^{x}
And the same idea that one can "undo" the other is still true:
ln(e^{x}) = x
e^{(ln x)} = x
And here are their graphs:
Natural Logarithm 
Natural Exponential Function 

Graph of f(x) = ln(x) 
Graph of f(x) = e^{x} 

Passes through (1,0) and (e,1) 
Passes through (0,1) and (1,e) 
They are the same curve with xaxis and yaxis flipped.
Which is another thing showing us they are inverse functions.
On a calculator the Natural Logarithm is the "ln" button. 
Always try to use Natural Logarithms and the Natural Exponential Function whenever possible.
The Common Logarithm
When the base is 10 we get:
 The Common Logarithm log_{10}(x), which is sometimes written as log(x)
Engineers love to use it, but it is not used much in mathematics.
On a calculator the Common Logarithm is the "log" button. It is handy because it tells us how "big" the number is in decimal (how many times we need to use 10 in a multiplication). 
Example: Calculate log_{10} 100
Well, 10 × 10 = 100, so when 10 is used 2 times in a multiplication we get 100:
log_{10} 100 = 2
Likewise log_{10} 1,000 = 3, log_{10} 10,000 = 4, and so on.
Example: Calculate log_{10} 369
OK, best to use my calculator's "log" button:
log_{10} 369 = 2.567...
Changing the Base
What if we want to change the base of a logarithm?
Easy! Just use this formula:
"x goes up, a goes down"
\frac{1}{log_{b} a} works as a "conversion factor" from one base to any other base.
Another useful property is:
log_{a} x = 1 / log_{x} a
See how "x" and "a" swap positions?
Example: Calculate 1 / log_{8} 2
1 / log_{8} 2 = log_{2} 8
And 2 × 2 × 2 = 8, so when 2 is used 3 times in a multiplication we get 8:
1 / log_{8} 2 = log_{2} 8 = 3
And we use the Natural Logarithm so often it is worth remembering this:
log_{a} x = ln x / ln a
Example: Calculate log_{4} 22
My calculator doesn't have a "log_{4}" button ... ... but it does have an "ln" button, so we can use that: 
What does this answer mean? It means that 4 with an exponent of 2.23 equals 22. So we can check that answer:
Check: 4^{2.23} = 22.01 (close enough!)
Here is another example:
Example: Calculate log_{5} 125
We can use the "ln" function on the calculator:
Is it exactly 3? We should not trust a calculator as there could be rounding errors, but in this case we can check that 5^{3} = 5 × 5 × 5 = 125 exactly, so:
Answer: 3
Real World Usage
Here are some uses for Logarithms in the real world:
Earthquakes
The magnitude of an earthquake is a Logarithmic scale.
The famous "Richter Scale" uses this formula:
M = log_{10} A + B
Where A is the amplitude (in mm) measured by the Seismograph
and B is a distance correction factor
Nowadays there are more complicated formulas, but they still use a logarithmic scale.
Sound
Loudness is measured in Decibels (dB for short):
Loudness in dB = 10 log_{10} (p × 10^{12})
where p is the sound pressure.
Acidic or Alkaline
Acidity (or Alkalinity) is measured in pH:
pH = −log_{10} [H^{+}]
where H^{+} is the molar concentration of dissolved hydrogen ions.
Note: in chemistry [ ] means molar concentration (moles per liter).
More Examples
Example: Solve 2 log_{8} x = log_{8} 16
But ... but ... but ... we can't have a log of a negative number!
So the −4 case is not defined.
Answer: 4
Check: use a calculator to see if this is the right answer ... also try the "−4" case.
Example: Solve e^{−w} = e^{2w+6}
Answer: w = −2
Check: e^{(−2)} = e^{2} and e^{2(−2)+6} = e^{2}
Footnote: Why does log(m × n) = log(m) + log(n) ?
To see why, we will use a^{loga(x)} = x and log_{a}(a^{x}) = x like this:So we seem to make things more complicated by transforming into a^{loga(x) }but then we are able to add them, then we transform back again and we have a solution!
It is one of those clever things we do in mathematics which can be described as "we can't do it here, so let's go over there, do it, then come back".