Logarithmic Functions

• Logarithmic Definition: A logarithm is defined as the power to which a fixed number, called the base ($a$), must be raised to produce a given number ($b$). This relationship is expressed as $\log_{a} b = x$, which is mathematically equivalent to the exponential form $a^x = b$.

• Base Constraints: For a logarithmic function to be well-defined, the base $a$ must be a positive constant ($a > 0$) and typically not equal to 1. The value being logged, known as the argument ($b$), must also be positive because a positive base raised to any real power will always yield a positive result.

• Inverse Nature: Logarithms and exponentials are inverse functions of one another. This means that applying a logarithm to its corresponding exponential function (or vice versa) 'undoes' the operation, resulting in the original variable: $\log_{a}(a^x) = x$ and $a^{\log_{a} x} = x$.

• The Logarithmic Question: When evaluating $\log_{a} b$, the central question being asked is: 'What power must I raise $a$ to in order to get $b$?' For example, $\log_{2} 8 = 3$ because $2^3 = 8$.

• Change of Form: Mastery of logarithms requires the ability to fluidly transition between exponential and logarithmic forms. This is the primary tool for isolating variables trapped in exponents.

• Monotonicity: Logarithmic functions are strictly increasing if the base $a > 1$ and strictly decreasing if $0 < a < 1$. This property ensures that every input has a unique output, making the function one-to-one.

Converting Forms

• Step 1: Identify the base ($a$), the exponent ($x$), and the result ($b$) in the equation $a^x = b$.
• Step 2: Rewrite the equation in the form $\log_{a} b = x$, ensuring the base of the exponent becomes the base of the logarithm.

Simplifying Expressions

• Inverse Property 1: Use $\log_{a}(a^k) = k$ to simplify logs where the argument is a power of the base. This is highly effective for mental math with common bases like 2, 3, or 10.
• Inverse Property 2: Use $a^{\log_{a} k} = k$ to simplify expressions where a base is raised to a logarithm of the same base. This effectively 'cancels' the logarithmic operation.

Solving Exponential Equations

• When an equation cannot be solved by equating bases (e.g., $2^x = 7$), apply a logarithm to both sides. This allows the exponent to be isolated and evaluated using a calculator.

• Argument vs. Logarithm Squaring: It is critical to distinguish between $\log(x^2)$ and $(\log x)^2$. The former applies the square to the input variable before logging, while the latter squares the entire numerical result of the logarithm.

• Exact Values: Examiners often require answers in 'exact form.' This means leaving the answer as $\ln 5$ or $\frac{\log 3}{\log 2}$ rather than providing a decimal approximation unless specifically asked.

• Hidden Quadratics: Look for patterns like $a^{2x} + p(a^x) + q = 0$. These can be solved by substituting $u = a^x$, solving the resulting quadratic for $u$, and then using logarithms to find the final values of $x$.

• Sanity Checks: Always verify that your final answer for $x$ does not result in taking the logarithm of a negative number in the original equation, as such solutions are undefined and must be rejected.