Properties of indices should be familiar to you from previous lessons. They are repeated here as introduction to using logarithms.
For indices, x and y \[ \begin{align} a^{x + y} &= a^x \times a^y \\ a^{x - y} &= \frac{a^x}{a^y} \\ (a^x)^y &= a^{xy} \\ &\quad \\ a^{-x} &= a^{-1x} = (\frac{1}{a})^x = \frac{1}{a^x} \end{align} \ \] And for bases a and b \[ \begin{align} (ab)^x &= a^xb^x \quad \\ (\frac{a}{b})^x &= \frac{a^x}{b^x} \\ \end{align} \ \]
When calculating with indices, the base often 'gets in the way'. In many cases it would be easier just to calculate directly with the indices. This is where logarithms come in.
A logarithm is the index of a number when the number is written as a power of the base.
\[ y = \log_a x \quad \text{means} \quad x = a^y \ \]
Note that we keep track of the base with the subscript on the log.
To find the value of loga x means to answer the question - to what power must a be raised to obtain x?
Index to log
Given a statement about an index you need to be able to make an equivalent statement about a log
Here is an easy way to remember this \[ \begin{align} \text{If } n &= b^i \\ \text{Then } \log_b n &= \log_b b^i \\ &= i \end{align} \ \]
Example \[ \begin{align} \text{If } 2^4 &= 16 \\ \text{Then } \log_2 16 &= \log_2 2^4 \\ &= 4 \end{align} \ \]
Log to index
Given a statement about a log you need to be able to make an equivalent statement about an index.
Here is an easy way to remember this \[ \begin{align} \text{If } \log_b n &= i \\ \text{Then } b^i &= n \\ \end{align} \ \]
Example \[ \begin{align} \text{If } \log_3 81 &= 4 \\ \text{Then } 3^4 &= 81 \\ \end{align} \ \]
Some properties of log are a direct consequence of the definition. \[ \begin{align} \log_a a &= \log_a a^1 \\ &= 1 \end{align} \ \] and \[ \begin{align} \log_a 1 &= \log_a a^0 \\ &= 0 \end{align} \ \]
To evaluate a log, use the definition.
Example To evaluate log2 32, first express 32 in terms of the base \[ \begin{align} \log_2 32 &= \log_2 2^5 \\ \end{align} \ \] Now the log is the power of 2 required to express 32 in base 2, so \[ \begin{align} \log_2 2^5 &= 5 \end{align} \ \]
Guided Examples
The exponential is a function
The exponential function is one to one. This means \[ \begin{align} \text{If } a^x &= a^y \\ \text{Then } x &= y \\ \end{align} \ \]
The logarithm is a function
The log of a negative number is not defined.
The logarithm function is one to one. This means \[ \begin{align} \text{If } \log_a x &= \log_a y \\ \text{Then } x &= y \\ \end{align} \ \]
The logarithm is the inverse function of the exponential.
Because the logarithm is the inverse of the exponential we have
\[ \begin{align} \log_a a^x &= x \quad \\ \end{align} \ \] because of the property of functions and their inverses: \[ \begin{align} f^{-1}(f(x)) &= x \quad \\ \end{align} \ \]