Comprehensive Logarithm Formula Sheet and Exponential Expression Formula Sheet

Product Rule: The logarithm of a product of two numbers, $x$ and $y$ , is equivalent to the sum of the logarithms of each individual factor, provided both have the same base $a$ . $\log_a(xy) = \log_a(x) + \log_a(y)$
Quotient Rule: The logarithm of a quotient (one number divided by another) is equal to the logarithm of the numerator minus the logarithm of the denominator. $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$
Power Rule: The logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of that number. $\log(x^n) = n\log(x)$

Logarithmic to Exponential Form: This rule defines the fundamental relationship between logs and exponents, stating that a logarithm of $b$ with base $a$ being equal to $c$ is logically equivalent to the base $a$ raised to the power of $c$ resulting in $b$ . $\log_a(b) = c \Leftrightarrow a^c = b$
Change of Base Formula: This formula allows for the conversion of a logarithm from one base to any other base. The logarithm of $b$ with base $a$ can be calculated by dividing the logarithm of $b$ by the logarithm of $a$ , using a new common base $c$ . $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$
Base Switch Rule: The logarithm of $b$ with base $a$ is equal to the reciprocal of the logarithm of $a$ with base $b$ . $\log_a(b) = \frac{1}{\log_b(a)}$

Identity Rule: When the base of the logarithm and the argument are identical, the value is always 1. $\log_a(a) = 1$
Zero Rule: Regardless of the base, the logarithm of 1 is always 0 because any non-zero base raised to the power of 0 equals 1. $\log(1) = 0$
Natural Logarithm of e: The natural logarithm (base $e$ ) of the mathematical constant $e$ is equal to 1. $\ln(e) = 1$

Logarithm Inverse Property: If the argument of a logarithm is the base of that logarithm raised to a power $n$ , the expression simplifies directly to that power. $\log_a(a^n) = n$
Inverse Exponent Property: If a base $a$ is raised to a power that is a logarithm with the same base $a$ and an argument $x$ , the result is simply the argument $x$ . $a^{\log_a(x)} = x$

Reciprocal Rules (Argument Reciprocal): The logarithm of the reciprocal of a number is the negative of the logarithm of that number. $\log_a\left(\frac{1}{b}\right) = -\log_a(b)$
Reciprocal Rules (Base Reciprocal): The logarithm of a number with a base that is a reciprocal (such as $\frac{1}{a}$ ) is equal to the negative of the logarithm of that same number with base $a$ . $\log_{\frac{1}{a}}(b) = -\log_a(b)$
Power with a Logarithmic Exponent: Also known as the base-exchange property, this rule states that when a number $a$ is raised to a logarithmic exponent with base $c$ , the base of the entire expression and the argument of the logarithm can be swapped. $a^{\log_c(b)} = b^{\log_c(a)}$
Rule 14 (Logarithm with Product Base): The logarithm of $a$ where the base is a product of $x$ and $y$ is equivalent to the reciprocal of the sum of the logarithms of $x$ and $y$ with base $a$ . $\log_{xy}(a) = \frac{1}{\log_a(x) + \log_a(y)}$
Rule 15 (Logarithm with Quotient Base): The logarithm of $a$ where the base is a quotient $\frac{x}{y}$ is equivalent to the reciprocal of the difference between the logarithms of $x$ and $y$ with base $a$ . $\log_{\frac{x}{y}}(a) = \frac{1}{\log_a(x) - \log_a(y)}$

Logarithm of Zero: The logarithmic function is not defined for an argument of zero; attempting to find the logarithm of zero results in an undefined value. $\log(0) = \text{Undefined}$
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