Exponentials & Logarithms

### Basic Rules of Exponents

Its worth remembering what "exponents" mean and that it is simply a short hand that saves a lot of writing… For example:

(1)
\begin{align} 4^3=4 \cdot 4 \cdot 4 \end{align}

Multiplying Exponents Given a expression such as $4^2 \cdot 4^3=4^5$. This can be seen by writing out the meaning of $4^2$ and $4^3$. Resulting in:

(2)
\begin{align} (4 \cdot 4) \cdot (4 \cdot 4 \cdot 4) = 4^5 \end{align}

Dividing Exponents Given an expression such as$\frac{3^4}{3^3}=3^1$. Again this can be seen by writing out the meaning of the numerator and the denominator:

(3)
\begin{align} \frac {3 \cdot 3 \cdot 3 \cdot 3 }{3 \cdot 3 \cdot 3}=3 \end{align}

Exponents of Exponents (Power Rule) Given an expression such $(2^4)^3=2^{12}$. This can be rewritten as:

(4)
\begin{align} (2^4) \cdot (2^4) \cdot (2^4)=2^{4+4+4}=2^{12} \end{align}

 General Rules $a^m \cdot a^n = a^{m+n}$ $\frac{a^m}{a^n}=a^{m-n}$ $(a^m)^n = a^{m \cdot n}$

Addition of Exponents Its worth noting that there are many times that exponents can not be simplified in particular with addition of two or more exponential terms such as:

(5)
$$4^3 + 3^4$$

The only time (in general) that addition can be simplified is if the base and and the exponent are the same such as:

(6)
$$3^2 + 3^2 = 2(3^2)$$

### Exponential Function

The basic exponential function looks something like:

(7)
$$f(x)=a^x$$

This function can be transformed in all the same ways as other functions. The graph of an exponential function looks like:

There are a few things worth noting about the graph. The function has a horizontal asymptote at $y=0$. That is the function gets closer and closer to the x-axis but never reaches it. This will always be true unless the graph is shifted vertically.

The y-intercept is also notable. Any number raised to the zero-power is 1 (i.e. $a^0=1$) the y-intercept (unless shifted horizontally or vertically) of a exponential function is always 1.

### Logarithmic Functions - Inverse Exponential Functions

The log function is a well known and feared function, but it is simply the inverse of a exponential function. It allows us to answer the question "3 raised to what power equals 27." Or in more mathy terms it lets us solve for x in expression like:

(8)
$$3^x = 27$$

Sure, most folk can figure out the one above, but it can get tricky such as:

(9)
$$2^x = 33,554,432$$

A minute or two with a calculator most folk can solve this one too, but wouldn't it be great to be a bit more clever? I think so, so does the IB…

The logarithm can be defined by:

If $a^x = b$ then $log_a(b)=x$

Essentially the logarithm "undoes" or "unwraps" the exponential function. Since the logarithm is simply the inverse of the exponential function then the graph of logarithm is related to the graph of the exponential function i.e. reflected across the line $y = x$

### Logarithm "Rules"

There are a few rules that are occasionally used in SL math. They aren't too complicated, but students often forget about them. These formula are in the formula booklet for exam starting in 2014. All of these rules can be proved some easier than others. For most folk the proofs aren't enlightening so I'll skip them.

#### Sum of Logarithms

If you have two logs added together of the same base the express can be simplified to the the log of the product of the arguments. Or in equation form:

(10)
$$log_c (a) + log_c(b) = log_c (ab)$$

#### Difference of Logarithms

If you have the difference of two logarithms of the same base the expression can be simplified to the log of the quotient of the arguments.

(11)
\begin{align} log_c (a) - log_c(b) = log_c \frac{a}{b} \end{align}

#### Power Rule

This rule is useful and the equation sort of speaks for itself - or at least I'm having a hard time describing it in simple words and phrases.

(12)
\begin{align} log_c a^r=r \cdot log_c a \end{align}

#### Change of Base

The first two rules require that both logarithms be the same base. That's not always true. So a change of base formula can be useful. It also used to be true that most GDC's could only do log's of base-10 or base-e. Most GDC's can now deal with any base or can be updated to do so.

(13)
\begin{align} log_b a = \frac {log_c a}{log_c b} \end{align}