Explain loga a clearly: what it is, why/how it happens, the key formula or law with a short worked example, and one common exam trap.

Step-by-step solution

1. 1

   Logarithm definition

   A logarithm answers: 'To what power must the base be raised to get the number?'

   $$log_a N = x \iff a^x = N$$
2. 2

   Apply definition to $log_a a$

   Let $log_a a = x$. We need to find the value of $x$.

   $$log_a a = x$$
3. 3

   Convert to exponential form

   Using the definition, $log_a a = x$ can be written as $a^x = a$.

   $$a^x = a$$
4. 4

   Solve for $x$

   Since $a = a^1$, we have $a^x = a^1$. Therefore, $x=1$.

   $$a^x = a^1 \implies x=1$$
5. 5

   Key formula

   Thus, $log_a a = 1$. This is a fundamental property of logarithms.

   $$log_a a = 1$$
6. 6

   Worked example

   Example: $log_5 5 = 1$ because $5^1 = 5$. Also, $log_{10} 10 = 1$ because $10^1 = 10$.

   $$log_5 5 = 1 \text{ (since } 5^1 = 5)$$
7. 7

   Exam trap

   Remember base conditions: $a > 0$ and $a \neq 1$. $log_1 1$ is undefined.

   $$\text{Base } a > 0, a \neq 1$$