Logarithmic Functions, Logarithmic Differentiation, and Inverse Trigonometric Functions

Logarithmic Functions

Definition and Properties

The logarithmic function is the inverse of the exponential function. For any base a where a > 0 and a ≠ 1, the logarithmic function with base a is defined as follows:

• Definition: $y = \log_a x$ if and only if $x = a^y$

• Function notation: $f(x) = \log_a x$ (read as "log base a of x")

• Inverse relationship: The logarithmic function is the inverse of the exponential function $a^x$.

Equivalent Logarithmic and Exponential Forms

Logarithmic and exponential equations can be written interchangeably:

Logarithmic Form	Exponential Form
$\log_2 8 = 3$	$2^3 = 8$
$\log_{1/2} 16 = -4$	$(\frac{1}{2})^{-4} = 16$
$\log_{10} 100,000 = 5$	$10^5 = 100,000$
$\log_3 \frac{1}{81} = -4$	$3^{-4} = \frac{1}{81}$
$\log_5 5 = 1$	$5^1 = 5$
$\log_{3/4} 1 = 0$	$(\frac{3}{4})^0 = 1$

General Logarithmic Function and Inverse Properties

• General form: $\log_a x = y \iff a^y = x$

• Inverse function property: If $f$ is one-to-one, then $f^{-1}$ is the inverse function of $f$ if and only if $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$ and $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$.

• Cancellation equations:

  • $\log_a(a^x) = x$ for every $x \in \mathbb{R}$

  • $a^{\log_a x} = x$ for every $x > 0$

Graphs of Exponential and Logarithmic Functions

Exponential and logarithmic functions are reflections of each other about the line $y = x$.

• Exponential function: $f(x) = 2^x$

• Logarithmic function: $g(x) = \log_2 x$

• Domain and Range:

  • For $f(x) = 2^x$: Domain $(-\infty, \infty)$, Range $(0, \infty)$

  • For $g(x) = \log_2 x$: Domain $(0, \infty)$, Range $(-\infty, \infty)$

• Key features of $y = \log_a x$ (for $a > 1$):

  • Domain: $(0, \infty)$

  • Range: $(-\infty, \infty)$

  • x-intercept: $(1, 0)$

  • Increasing function

  • One-to-one (has an inverse)

  • Vertical asymptote at $x = 0$

  • Continuous

  • Reflection of $y = a^x$ about $y = x$

Properties of Logarithms

• Inverse Properties:

  • $a^0 = 1$ because $a^0 = 1$

  • $\log_a 1 = 0$ because $a^0 = 1$

  • $\log_a a = 1$ because $a^1 = a$

  • $a^{\log_a x} = x$ and $\log_a a^x = x$

  • If $\log_a x = \log_a y$, then $x = y$ (One-to-One Property)

• Laws of Logarithms:

  • Product Law: $\log_a (AB) = \log_a A + \log_a B$

  • Quotient Law: $\log_a \left(\frac{A}{B}\right) = \log_a A - \log_a B$

  • Power Law: $\log_a (A^C) = C \log_a A$

Expanding and Combining Logarithmic Expressions

• Expanding: Use the laws of logarithms to write expressions as sums, differences, or multiples of logs.

  • Example: $\log_2 6x = \log_2 6 + \log_2 x$

  • Example: $\log_4 \frac{z^2}{y} = 2\log_4 z - \log_4 y$

  • Example: $\log_5 x^3 y^6 = 3\log_5 x + 6\log_5 y$

  • Example: $\log \frac{ab}{\sqrt[3]{c}} = \log a + \log b - \frac{1}{3} \log c$

• Combining: Use the laws in reverse to write sums or differences as a single logarithm.

  • Example: $3\log x + 2\log(x-5) = \log(x^3) + \log((x-5)^2) = \log(x^3(x-5)^2)$

  • Example: $3\log s - \frac{1}{2}\log(t+1) = \log(s^3) - \log((t+1)^{1/2}) = \log\left(\frac{s^3}{\sqrt{t+1}}\right)$

• Common Logarithm: For all positive numbers $x$, $\log x = \log_{10} x$

Logarithmic Differentiation

Concept and Application

Logarithmic differentiation is a technique used to find the derivative of complicated functions, especially those involving products, quotients, or powers. The process involves taking the natural logarithm of both sides of an equation and then differentiating implicitly.

• Take natural logarithms of both sides: $y = f(x) \implies \ln y = \ln f(x)$

• Use properties of logarithms to simplify $\ln f(x)$

• Differentiate both sides with respect to $x$

• Solve for $y'$

Example: Find $\frac{dy}{dx}$ if $y = x^x$ for $x > 0$.

• Take $\ln$ of both sides: $\ln y = x \ln x$

• Differentiate: $\frac{1}{y} \frac{dy}{dx} = \ln x + 1$

• So, $\frac{dy}{dx} = y(\ln x + 1) = x^x (\ln x + 1)$

Inverse Trigonometric Functions

Definition and Properties

Inverse trigonometric functions are defined as the inverses of the restricted trigonometric functions. Since trigonometric functions are not one-to-one over their entire domains, their domains are restricted to make them invertible.

• Arcsine: $y = \sin^{-1} x$ is the number in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ for which $\sin y = x$

• Arccosine: $y = \cos^{-1} x$ is the number in $[0, \pi]$ for which $\cos y = x$

• Arctangent: $y = \tan^{-1} x$ is the number in $(-\frac{\pi}{2}, \frac{\pi}{2})$ for which $\tan y = x$

Domains and Ranges

Function	Domain	Range
$y = \sin^{-1} x$	$[-1, 1]$	$[-\frac{\pi}{2}, \frac{\pi}{2}]$
$y = \cos^{-1} x$	$[-1, 1]$	$[0, \pi]$
$y = \tan^{-1} x$	$(-\infty, \infty)$	$(-\frac{\pi}{2}, \frac{\pi}{2})$

Properties and Identities

• $\sin(\sin^{-1} x) = x$ for $-1 \leq x \leq 1$

• $\cos(\cos^{-1} x) = x$ for $-1 \leq x \leq 1$

• $\tan(\tan^{-1} x) = x$ for all $x \in \mathbb{R}$

• Cofunction identities:

  • $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$

  • $\cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x$

Examples

• Evaluate $\sin^{-1}(\frac{\sqrt{3}}{2})$:

  • $\sin^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{3}$ because $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$

• Evaluate $\cos^{-1}(-\frac{1}{2})$:

  • $\cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3}$ because $\cos(\frac{2\pi}{3}) = -\frac{1}{2}$

Derivatives of Inverse Trigonometric Functions

• $\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}$ for $|x| < 1$

• $\frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}}$ for $|x| < 1$

• $\frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2}$ for all $x$

Example: Find $\frac{d}{dx} \sin^{-1}(e^x)$:

• $\frac{d}{dx} \sin^{-1}(e^x) = \frac{e^x}{\sqrt{1 - (e^x)^2}}$

Summary Table: Laws of Logarithms

Law	Formula	Explanation
Product	$\log_a (AB) = \log_a A + \log_a B$	Logarithm of a product is the sum of the logarithms.
Quotient	$\log_a \left(\frac{A}{B}\right) = \log_a A - \log_a B$	Logarithm of a quotient is the difference of the logarithms.
Power	$\log_a (A^C) = C \log_a A$	Logarithm of a power is the exponent times the logarithm of the base.

Additional info: These notes cover foundational concepts in calculus related to logarithmic and exponential functions, their properties, differentiation techniques, and inverse trigonometric functions, which are essential for understanding advanced calculus topics such as integration and solving differential equations.