Laws of Logarithms for University Admission

Logarithms are very closely related to indices. In fact, they are really the inverse of indices: they tell you what power a number has to be raised to, rather than raising a number to a power. Historically, logarithms were invented to make complex calculations easier before the existence of calculators, by converting multiplication into addition.

The Fundamental Definition

The base of a logarithm, written as a subscript, identifies the number being raised to a power. For example, $\log_{10}$ tells you what power 10 needs to be raised to in order to get a given number:

1. $\log_{10} 10 = 1$ because $10^1 = 10$.
2. $\log_{10} 100 = 2$ because $10^2 = 100$.
3. $\log_{10} 1000 = 3$ because $10^3 = 1000$.
4. $\log_{10} 27 \approx 1.43136$ because $10^{1.43136\dots} = 27$.

We can use any base $a$. For example, $\log_2$ tells you what power 2 needs to be raised to reach a value:

1. $\log_2 32 = 5$ because $2^5 = 32$.
2. $\log_2 \frac{1}{2} = -1$ because $2^{-1} = \frac{1}{2}$.

In general, the relationship is $a^b = c \Leftrightarrow b = \log_a c$. There are three critical restrictions to remember for the TMUA: the base $a$ must be positive ($a > 0$) and not equal to 1, the number we take the log of ($c$) must be positive ($c > 0$), but the resulting log ($b$) can be any real number, including zero or negative values.

Graphical Representation

We can understand logarithms by looking at them as the inverse of exponential graphs. Consider $y = 2^x$. This function takes an input from the $x$ axis and provides a $y$ value. If we start on the $y$ axis and trace back to the $x$ axis, we are finding the logarithm.

The graph of $y = \log_2 x$ is simply the graph of $y = 2^x$ with the $x$ and $y$ axes swapped. This is a reflection in the line $y = x$.

Note that the log graph is only defined for $x > 0$ and it always crosses the $x$ axis at 1, because $a^0 = 1$ for any base $a$, meaning $\log_a 1 = 0$.

The Laws of Logarithms

You must be fluent in manipulating logarithms using these four primary rules. Their derivations are rooted in the laws of indices.

1. The Product Rule: $\log_a x + \log_a y = \log_a(xy)$ This is the logarithmic equivalent of $a^p a^q = a^{p+q}$. We can see this works because $a^{\log_a x + \log_a y} = a^{\log_a x} a^{\log_a y} = xy = a^{\log_a(xy)}$.

2. The Quotient Rule: $\log_a x - \log_a y = \log_a (\frac{x}{y})$ This corresponds to the index law $\frac{a^p}{a^q} = a^{p-q}$. Derivation: $a^{\log_a x - \log_a y} = \frac{a^{\log_a x}}{a^{\log_a y}} = \frac{x}{y} = a^{\log_a (\frac{x}{y})}$.

3. The Power Rule: $k \log_a x = \log_a (x^k)$ This follows from $(a^p)^k = a^{pk}$. Derivation: $a^{k \log_a x} = (a^{\log_a x})^k = x^k = a^{\log_a (x^k)}$.

4. Special Cases:

• The Reciprocal Rule: $\log_a (\frac{1}{x}) = -\log_a x$. This is a specific instance of the power rule where $k = -1$.
• Base Identity: $\log_a a = 1$, because $a^1 = a$.

Change of Base Formula

Although questions specifically requiring the change of base formula will not be set in the TMUA, it is a useful tool for your mathematical kit. It allows you to convert a logarithm from one base to another:

$\log_a b = \frac{\log_c b}{\log_c a}$

A useful result of this is that $\log_a b = \frac{1}{\log_b a}$. You can derive this by letting $p = \log_a b$, which means $a^p = b$. Taking $\log_c$ of both sides gives $\log_c a^p = \log_c b$, or $p \log_c a = \log_c b$, leading to $p = \frac{\log_c b}{\log_c a}$.