Laws of Logarithms

• A logarithm is defined by the relationship $a^x = b \iff \log_a b = x$, where $a$ is the base, $x$ is the exponent, and $b$ is the result. This means the logarithm is essentially the power you raise the base to in order to get the value $b$.

• The base $a$ must be a positive constant ($a > 0$) and not equal to 1, while the argument $b$ must always be greater than zero. Logarithms of negative numbers or zero are undefined in the real number system.

• Natural Logarithms use the mathematical constant $e$ (approximately 2.718) as the base and are denoted as $\ln x$, which is equivalent to $\log_e x$.

• The Product Law: $\log_a(xy) = \log_a x + \log_a y$. This principle states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, mirroring the index law $a^m \cdot a^n = a^{m+n}$.

• The Quotient Law: $\log_a(\frac{x}{y}) = \log_a x - \log_a y$. This rule dictates that the logarithm of a division is the difference between the logarithm of the numerator and the denominator, corresponding to the index law $\frac{a^m}{a^n} = a^{m-n}$.

• The Power Law: $\log_a(x^k) = k \log_a x$. This allows an exponent within a logarithm to be moved to the front as a multiplier, which is the primary mechanism used to solve for unknown variables in exponents.

Solving Exponential Equations

• Step 1: Take Logarithms: Apply $\ln$ or $\log_{10}$ to both sides of the equation (e.g., $a^x = b$ becomes $\ln(a^x) = \ln b$).

• Step 2: Apply Power Law: Move the variable exponent to the front of the logarithm (e.g., $x \ln a = \ln b$).

• Step 3: Isolate the Variable: Divide by the logarithmic constant to solve for $x$ (e.g., $x = \frac{\ln b}{\ln a}$).

Handling Hidden Quadratics

• Equations in the form $p(a^{2x}) + q(a^x) + r = 0$ can be treated as quadratics by substituting $u = a^x$, resulting in $pu^2 + qu + r = 0$. After solving for $u$, you must then solve the resulting exponential equations for $x$.

• It is vital to distinguish between $\log(x^2)$, where the power law applies, and $(\log x)^2$, which is the entire logarithmic value squared and cannot be simplified using the power law.