Laws of Logarithms

• A logarithm is defined as the power to which a fixed number, called the base ($a$), must be raised to produce a given number ($b$). The relationship is expressed as $a^x = b \iff \log_a b = x$.

• The Natural Logarithm ($\ln x$) is a specific case where the base is the mathematical constant $e$ (approximately $2.718$). It is written as $\ln x \equiv \log_e x$.

• Logarithms are only defined for positive values of the argument ($b > 0$) and positive bases ($a > 0, a \neq 1$). This is because raising a positive base to any real power always results in a positive value.

• The Multiplication Law: The logarithm of a product is equal to the sum of the logarithms of the factors. This is expressed as $\log_a(xy) = \log_a x + \log_a y$.

• The Division Law: The logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator. This is expressed as $\log_a(\frac{x}{y}) = \log_a x - \log_a y$.

• The Power Law: The logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number. This is expressed as $\log_a(x^k) = k \log_a x$.

• The laws of logarithms are direct consequences of the laws of indices. For example, the multiplication law for logs corresponds to the index law $a^m \times a^n = a^{m+n}$.

• Because logarithms and exponentials are inverse functions, they 'undo' each other. This leads to the identities $\log_a(a^x) = x$ and $a^{\log_a x} = x$.

• A critical identity derived from the power law is $\log_a 1 = 0$, which stems from the fact that any non-zero base raised to the power of $0$ equals $1$ ($a^0 = 1$).

Solving Exponential Equations

• Step 1: Isolate the exponential term. Ensure the equation is in the form $a^{f(x)} = c$ or $a^{f(x)} = b^{g(x)}$.
• Step 2: Take logarithms of both sides. Usually, the natural logarithm ($\ln$) is used for convenience.
• Step 3: Apply the Power Law. Move the exponents to the front as coefficients: $f(x) \ln a = \ln c$.
• Step 4: Solve for the variable. Rearrange the resulting linear or quadratic equation to find $x$.

Handling Hidden Quadratics

• Equations like $e^{2x} + 5e^x + 6 = 0$ can be solved by using a substitution such as $u = e^x$. This transforms the equation into a standard quadratic form $u^2 + 5u + 6 = 0$.

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