Appendix D — Appendix: Exponents & Logarithms

D.1 Exponents

Exponents are of the form a^x where:

a (called the base) and
x (called the exponent) is any real number.

Recall that a^0 = 1. Exponents have the following useful properties

a^x· a^y = a^{x+y}
(a^x)^y = a^{x·y}

Note: that the first property requires that both terms have the same base a.

We cannot simplify a^x ·b^y if a \ne b.

One common base is the number e which is approximately equal to 2.7183.
The function e^x is so common in mathematics and has its own symbol e^x = \exp(x).
Because e > 0 we have e^x > 0 for all real numbers x
\lim_{x \to \infty} x = e^{−x} = 0.

D.2 Natural Logarithms

We will need to manipulate long products of probabilities. Since there often comprise small fractions, their calculation on computers can be problematic due to the underflow of floats. We will therefore prefer to convert these products into sums of logarithms.

Definition D.1 (The Logarithm) A log is the inverse of a power. We can use (Equation D.1).

y = a^x \implies log_a(y) = x \tag{D.1}

Definition D.2 (The Natural log) The natural logarithm function has base e and is written without the subscript

log_e(y) = log(y) \tag{D.2}

Theorem D.1 (Logs take positive values) logs only exist for values greater than 0

\forall x(e^x > 0) \implies \exists \log(y) \iff {y > 0}

We can use the properties of exponents from the previous section to obtain some important properties of logarithms:

Definition D.3 (Log of a product) we can use Equation D.3 to convert a log of a product to a sum of logs.

\log(x·y) = \log(x) + \log(y) \tag{D.3}

Definition D.4 (Log of a quotient) we can use Equation D.4 to convert a log of a quotient to a difference of logs.

\log(\frac{x}{y}) = log(x) − log(y) \tag{D.4}

Definition D.5 (Log of a power) we can use Equation D.5 to convert a log of a variable raised to a power into the product.

\log(x^b) = b \cdot log(x) \tag{D.5}

Definition D.6 (Log of one) we can use (Equation D.6) to replace a log of 1 with zero since $x(x^0 = 1) $

\log(1)=0 \tag{D.6}

D.2.1 Log of exponent

we can use (Equation D.7) to cancel a log of an exponent since the log is the inverse function of the exponent.

exp(log(y)) = log(exp(y)) = y \tag{D.7}

Example D.1 (Logarithm) \begin{aligned} log \frac{5^2}{10}= 2 log(5) − log(10) ≈ 0.916. \end{aligned}

Definition D.7 (Change of base for a log) we can use (Equation D.8) to change the base of a logarithm.

\log_b(a)=\frac{\log_c(a)}{\log_c(n)} \tag{D.8}

Definition D.8 (Derivative of a Log) we can use (Equation D.9) to differentiate a log.

\frac{d}{dx} \log_(x)=\frac{1}{x} \tag{D.9}

Because the natural logarithm is a monotonically increasing one-to-one function, finding the x which maximizes any (positive-valued function) f(x) is equivalent to maximizing log(f(x)).
This is useful because we often take derivatives to maximize functions.
If f(x) has product terms, then log(f(x)) will have summation terms, which are usually simpler when taking derivatives.

--- title: "Appendix: Exponents & Logarithms" --- ## Exponents {#sec-exponents} Exponents are of the form $a^x$ where: - a (called the base) and - x (called the exponent) is any real number. Recall that $a^0 = 1$. Exponents have the following useful properties 1. $a^x· a^y = a^{x+y}$ 2. $(a^x)^y = a^{x·y}$ Note: that the first property requires that both terms have the same base a. We cannot simplify $a^x ·b^y$ if $a \ne b$. - One common base is the number e which is approximately equal to 2.7183. - The function $e^x$ is so common in mathematics and has its own symbol $e^x = \exp(x)$. - Because $e > 0$ we have $e^x > 0$ for all real numbers x - $\lim_{x \to \infty} x = e^{−x} = 0$. ## Natural Logarithms {#sec-natural-logarithms} We will need to manipulate long products of probabilities. Since there often comprise small fractions, their calculation on computers can be problematic due to the underflow of floats. We will therefore prefer to convert these products into sums of logarithms. ::: {#def-log-exp} ### The Logarithm A log is the inverse of a power. We can use (@eq-log-def). $$ y = a^x \implies log_a(y) = x $$ {#eq-log-def} ::: ::: {#def-natural-log} ### The Natural log The natural logarithm function has base e and is written without the subscript $$ log_e(y) = log(y) $$ {#eq-ln-def} ::: ::: {#thm-log-positive} ### Logs take positive values logs only exist for values greater than 0 $$ \forall x(e^x > 0) \implies \exists \log(y) \iff {y > 0} $$ ::: We can use the properties of exponents from the previous section to obtain some important properties of logarithms: ::: {#def-log-product} ### Log of a product we can use @eq-log-product to convert a log of a product to a sum of logs. $$ \log(x·y) = \log(x) + \log(y) $$ {#eq-log-product} ::: ::: {#def-log-quotient} ### Log of a quotient we can use @eq-log-quotient to convert a log of a quotient to a difference of logs. $$ \log(\frac{x}{y}) = log(x) − log(y) $$ {#eq-log-quotient} ::: ::: {#def-log-power} ### Log of a power we can use @eq-log-power to convert a log of a variable raised to a power into the product. $$ \log(x^b) = b \cdot log(x) $$ {#eq-log-power} ::: ::: {#def-log-one} ### Log of one we can use (@eq-log-one) to replace a log of 1 with zero since $\forall x(x^0 = 1) $ $$ \log(1)=0 $$ {#eq-log-one} ::: ### Log of exponent we can use (@eq-log-exp) to cancel a log of an exponent since the log is the inverse function of the exponent. $$ exp(log(y)) = log(exp(y)) = y $$ {#eq-log-exp} ::: {#exm-log} ### Logarithm $$ \begin{aligned} log \frac{5^2}{10}= 2 log(5) − log(10) ≈ 0.916. \end{aligned} $$ ::: ::: {#def-log-bases} ### Change of base for a log we can use (@eq-log-bases) to change the base of a logarithm. $$ \log_b(a)=\frac{\log_c(a)}{\log_c(n)} $$ {#eq-log-bases} ::: ::: {#def-log-derivative} ### Derivative of a Log we can use (@eq-log-derivative) to differentiate a log. $$ \frac{d}{dx} \log_(x)=\frac{1}{x} $$ {#eq-log-derivative} ::: - Because the natural logarithm is a monotonically increasing one-to-one function, finding the x which maximizes any (positive-valued function) $f(x)$ is equivalent to maximizing $log(f(x))$. - This is useful because we often take derivatives to maximize functions. - If $f(x)$ has product terms, then $log(f(x))$ will have summation terms, which are usually simpler when taking derivatives.