Algebra - Logarithms

Definition of Logarithm: A logarithm is the inverse operation of exponentiation. If $a^x = y$, then we say that the logarithm of $y$ to the base $a$ is $x$, written as $\log_a y = x$. This relationship connects the base $a$, the exponent $x$, and the resulting value $y$. Visually, if you imagine an exponential growth curve rising steeply, the logarithm reflects this across the line $y=x$, showing how the exponent grows slowly as the value increases.

Existence Conditions: For a logarithm $\log_a m$ to be defined, the base $a$ must be a positive real number not equal to $1$ ($a > 0, a \neq 1$), and the number $m$ must be positive ($m > 0$). On a coordinate plane, the graph of $y = \log_a x$ only exists to the right of the y-axis, indicating that we cannot take the logarithm of zero or negative numbers.

Fundamental Identities: There are two critical values to remember. First, $\log_a a = 1$ because any number raised to the power of $1$ is itself ($a^1 = a$). Second, $\log_a 1 = 0$ because any non-zero number raised to the power of $0$ is $1$ ($a^0 = 1$). This means the graph of every logarithmic function $y = \log_a x$ passes through the point $(1, 0)$ on the x-axis.

Product and Quotient Rules: Logarithms convert multiplication into addition and division into subtraction. Specifically, the log of a product is the sum of the logs: $\log_a (mn) = \log_a m + \log_a n$. Conversely, the log of a quotient is the difference of the logs: $\log_a (\frac{m}{n}) = \log_a m - \log_a n$. This effectively 'compresses' operations, making large-scale calculations easier to handle.

Power Rule: If the argument of a logarithm has an exponent, that exponent can be moved to the front of the logarithm as a multiplier: $\log_a (m^n) = n \log_a m$. Visually, this shows that scaling the exponent of the input results in a linear scaling of the output value.

Common Logarithms: Logarithms with base $10$ are known as common logarithms. In many textbooks and calculators, if no base is specified (e.g., $\log x$), the base is assumed to be $10$. These are particularly useful for expressing numbers in scientific notation and calculating magnitudes.

Base Change and Reciprocal Property: The value of a logarithm can be expressed using a different base using the formula $\log_a m = \frac{\log_b m}{\log_b a}$. Additionally, swapping the base and the argument results in a reciprocal: $\log_a b = \frac{1}{\log_b a}$.