Logarithm Laws

If you are studying logarithms it is reasonably safe to assume that you’re already reasonably familiar with index rules; those shortcuts that allow for swift calculation of exponents when dealing with equal bases. The log laws are really just a rearrangement of these.

First, multiplying. Since when you multiply terms with the same base you add the indices, the log of a product is equal to the sum of the logs of the factors of that product. This is best demonstrated starting with the index rule and working through an example.

\begin {aligned}a^m \times a^n &= a^{m+n} \\ 8 \times 16 &= 128 \\2^3 \times 2^4 &= 2^7 \\ \log_2{8} + \log_2{16} &= \log_2{128} \\ \log_a{m} + \log_a{n} &= \log_a{mn}\end{aligned}Index / log law for multiplication / addition

Second, division. As the inverse of multiplication this one is now hopefully easy to spot: subtracting the parts. The log of a quotient is equal to the difference between the logs of the dividend and the divisor (numerator and denominator).

\begin {aligned}\frac{a^m}{a^n} &= a^{m-n} \\ \log{\frac{m}{n}}&= \log{m} - \log{n}\end{aligned}Index / log law for division / subtraction

Using this rule for division it follows that we should be able to find the value of a logarithm of 1. As any value divided by itself is 1, choosing a value expressed in index form allows us to apply the previous rule.

\begin {aligned}\frac{a^b}{a^b} &= 1 \\ a^{b-b} &= 1 \\ \log_a{1} &= b-b \\ \log_a{1} &= 0 \end {aligned}The log of 1 in any base is 0

Terms with indices raised to a power are an extension of the multiplication of indices already described. This approach can be applied to the addition of logarithms:

\begin {aligned}(a^m)^n &= a^m \times a^m \times \ldots \times a^m \\ \log_a{m^n} &= \log_a{m} + \log_a{m} + \ldots + \log_a{m} \\ \log_a{m^n} &= n \log_a{m}\end {aligned}Powers in logarithms

Differentiation – Introduction

Differentiation is the process we use to find an expression for the rate of change of a function. This is most easily explained graphically.

In the graphs above, the green curve shows a function with a point moving along it. The tangent at that (moving) point is shown. The blue curve shows the value of the gradient of that tangent as x changes.

By drawing tangents to a curve we can estimate the gradient at any point but this is time consuming for more than a couple of points, requires a well-drawn graph, and is inaccurate. Differentiation allows us to find a function that describes all of these tangents without having to draw them.

The result of differentiating is the derivative of the function. Because a derivative describes how one value is affected by a change in a variable we say that we find the derivative with respect to that variable. Differentiating the function that gives the green curve above results in the function that gives the blue curve. The blue curve is the gradient function of the green one.

For a graph of y against x the gradient is the derivative of y with respect to x, this is written as \frac{\mathrm{d}y}{\mathrm{d}x}. Using function notation, \frac{\mathrm{d}y}{\mathrm{d}x}=f'(x).

There are numerous applications of derivatives. One of the most easily understood is velocity. Velocity is the rate of change of displacement with respect to time, often written v=\frac{\mathrm{d}x}{\mathrm{d}t}=\dot{x}.

Logarithms

Logarithms are a way of expressing powers.

The logarithm \log_b{x} for a base b and a number x is defined to be the inverse function of taking b to the power x, i.e., b^x.¹

The simplest way to explain the concept of logarithms is through an example:

\begin{aligned}2^5&=32 \\ \log_2{32}&=5\end {aligned}

The subscript of the logarithm above denotes the base of the logarithm. This is equivalent to the number which is raised to a power in the preceding equation. The logarithm to the base 2 of 32 is 5, because 2 to the power of 5 is 32.

The equation \log_3{81}=x can be solved by asking ourselves “what power of 3 gives a result of 81?” In this case the answer will be x=4 and can be calculated relatively easily. For most questions involving logarithms you will be resorting to the functions in your calculator or spreadsheet (or log tables, which have become increasingly rare).

x^y = z \iff \log_x{z} = y

¹ Weisstein, Eric W. “Logarithm.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Logarithm.html