Logarithms
Logarithms are the inverses of exponentials.
Graphs
In general, exponential functions look like this:
On the other hand, logarithms look like this:
Syntax
𝑦 = 𝑏 𝑥 is equivalent to log 𝑏 𝑦 = 𝑥.
In this form, and throughout this document, we will refer to b as the “base” of our logarithms.
Types of Logarithms

If b=10, you are dealing with a common logarithm or just a common log for short. If you
ever see a logarithm without a base (simply written like log 𝑦), you are dealing with a
common log.

If b=e, you are dealing with a natural logarithm. Unlike other logs, they are written
completely different!
ln 𝑦
(Compare this with log 𝑒 𝑦.)
The number e is much like π! It is a mathematical constant estimated at around 2.71828,
but it’s really infinite. As of 2010, we know up to 1,000,000,000,000 digits of e! Pretty
neat.
Properties of Logarithms
log 𝑏 (𝑥) + log 𝑏 (𝑦)
log 𝑏 (𝑥 × 𝑦)
𝑥
log 𝑏 ( )
𝑦
log 𝑏 (𝑥 𝑦 )
log 𝑏 (𝑥) − log 𝑏 (𝑦)
y × log 𝑏 (𝑥)
log 𝑥
log 𝑏
log 𝑏 (𝑥)
This is called the change of base property.
Notice how it changes your log to two
common logs. If you have an older calculator
that doesn’t allow you to compute anything
but a common log, use this property!
Doman Shifts of Logarithms
Consider log 𝑏
𝑦 = 𝑥.
y can NEVER be zero or negative! Use this fact to find your domain.
Example:
Find the domain of log(3𝑥 + 4).
In this example, 3𝑥 − 4 must be non-zero and positive. Use an inequality to find
where x lives.
3𝑥 + 4 > 0
3𝑥 > −4
𝑥>
−4
3
4
Therefore, the domain of this function is (− , ∞).
3
4
There are rounded parenthesis on − because if you plug it into your logarithm,
3
you’ll get a zero inside the log which can never happen!
Values of Logarithms You Should Know

log 𝑏 1 = 0
o This applies for any base log.
o Even e! ln(1) = 0.


log 𝑏 𝑏 = 1
o log(10) = 1
o ln(𝑒) = 1
Log of 10s
o log(10) = 1
o log(100) = 2
o log(1000) = 3
o Notice the pattern? The number of zeros in your logarithm! You can also use a
property to reason this out.
4
log(10000) = log (10 ) = 4 × log(10) = 4 × 1 = 1
Other Examples
Example:
Solve for x.
ln(4𝑥 + 9) = 𝑥 2 + 9
Recall that 𝑦
= 𝑏 𝑥 is equivalent to log 𝑏 𝑦 = 𝑥.
So, in the case of this problem, log 𝑒 (4𝑥 + 9)
9 = 𝑒𝑥
2 +9
= 𝑥2 + 9 is equivalent to 4𝑥 +
.
You can also think of it as raising e by both sides of the equality. If this were a
common log, raise 10 by both sides of the equality.
Example:
Solve for x.
log(3𝑥 + 4) = 1
Let’s do what I said in the previous example and raise 10 by both sides of the
equality.
10log(3𝑥+4) = 101
3𝑥 + 4 = 10
3𝑥 = 6
𝑥=2
Example:
Solve for Q.
𝐸 = 𝐸° −
𝑅𝑇
× ln 𝑄
𝑛𝐹
(Look familiar? It’s the Nernst equation from Chem 112!)
This is an ugly equation with a really ugly result, but if you can handle this problem,
there isn’t must you can’t do!
The first step is to get ln 𝑄 alone on one side.
𝐸 − 𝐸° = −
𝑅𝑇
× ln 𝑄
𝑛𝐹
𝐸 − 𝐸°
= ln 𝑄
𝑅𝑇
−
𝑛𝐹
𝑛𝐹 × (𝐸 ° − 𝐸)
= ln 𝑄
𝑅𝑇
Getting lost with the algebra? No worries—come see us in the Math Dimension!
Take the “e” of both sides.
𝑄=
𝑛𝐹×(𝐸 ° −𝐸)
𝑒 𝑅𝑇