In-depth Notes on Logarithmic, Exponential, and Hyperbolic Functions

Section 7.1 Logarithmic and Exponential Functions Revisited

  • Definition of Natural Logarithm: The natural logarithm ( \ln(x) ) of a number ( x > 0 ) is defined based on the integral of 1/t from 1 to x, yielding:

  • If ( x > 1 ) then ( \ln(x) > 0 )

  • If ( 0 < x < 1 ) then ( \ln(x) < 0 )

  • Properties of the Natural Logarithm (Theorem 7.1):

  • Domain: (0, ( \infty ))

  • Range: (−∞, ( \infty ))

  • The function ( y = \ln(x) ) is increasing for ( x > 0 )

  • It is concave down for all ( x > 0 )

The Number e

  • Definition: The number ( e ) is defined as the limit of (1 + 1/n)ⁿ as n approaches infinity. It approximates to 2.718281828…

  • Natural Logarithm of e: ( \ln(e) = 1 )

  • Example area under curve shows ( \ln(e) ) related to the integral of 1/t from 1 to e.

The Exponential Function

  • Definition: The exponential function ( e^x ) is defined for any real number x.

  • Properties of Exponential Functions (Theorem 7.2):

  1. ( e^{x+y} = e^x \cdot e^y )

  2. ( e^{0} = 1 )

  3. The function grows rapidly for large x.

  4. ( \frac{d}{dx}(e^x) = e^x )

  5. ( \int e^x dx = e^x + C )

Generalized Exponential Functions

  • For any base ( b ) (where ( b > 0 ) and ( b \neq 1 )), the exponential function is expressed as ( b^x = e^{x \ln(b)} ).

Exponential Growth

  • Growth Function: Defined as ( y = y0 e^{kt} ), where ( y0 ) is initial value and k is growth constant (k > 0).

  • Doubling Time: The time required for a quantity to double, given by the formula ( T_d = \frac{
    \ln(2)}{k} ).

  • Examples: Exponential functions in population growth, resource consumption, etc.

Exponential Decay

  • Decay Function: Expressed as ( y = y_0 e^{-kt} ), where k < 0.

  • Half-Life: The time taken for a quantity to reduce to half its initial value, given by the formula ( T_{1/2} = \frac{\ln(2)}{-k} ).

Section 7.3 Hyperbolic Functions

  • Definitions:

  • ( \cosh(x) = \frac{e^x + e^{-x}}{2} )

  • ( \sinh(x) = \frac{e^x - e^{-x}}{2} )

  • Properties:

  • Hyperbolic sine is an odd function, whereas hyperbolic cosine is even.

  • Derivatives:

    • ( \frac{d}{dx} \sinh(x) = \cosh(x) )

    • ( \frac{d}{dx} \cosh(x) = \sinh(x) )

  • Hyperbolic Identities:

  • ( \cosh^2(x) - \sinh^2(x) = 1 )

  • Addition formulas for hyperbolic functions

  • Graphs: Each hyperbolic function has a characteristic shape with defined domains and ranges.

  • Inverse Hyperbolic Functions: Represented in terms of logarithms, which can be found using definitions and properties derived from the original hyperbolic functions.

Mathematical Formulas

Natural Logarithm

  • ( \ln(x) \text{ for } x > 0 )

  • If ( x > 1 ): ( \ln(x) > 0 )

  • If ( 0 < x < 1 ): ( \ln(x) < 0 )

Properties of Natural Logarithm

  • Domain: (0, ( \infty ))

  • Range: (−( \infty ), ( \infty ))

  • ( y = \ln(x) \text{ is increasing for } x > 0 )

  • Concave down for all ( x > 0 )

The Number e

  • ( e = \lim_{n\to\infty}(1 + \frac{1}{n})^n \approx 2.718281828… )

  • ( \ln(e) = 1 )

Exponential Function

  • ( y = e^x )

  • Properties:

  1. ( e^{x+y} = e^x \cdot e^y )

  2. ( e^0 = 1 )

  3. Growth property: rapidly increases for large ( x )

  4. ( \frac{d}{dx}(e^x) = e^x )

  5. ( \int e^x dx = e^x + C )

Generalized Exponential Functions

  • ( b^x = e^{x \ln(b)} \text{ for } b > 0, b \neq 1 )

Exponential Growth

  • ( y = y_0 e^{kt} )

  • ( T_d = \frac{\ln(2)}{k} )

Exponential Decay

  • ( y = y_0 e^{-kt} )

  • ( T_{\frac{1}{2}} = \frac{\ln(2)}{-k} )

Hyperbolic Functions

  • ( \cosh(x) = \frac{e^x + e^{-x}}{2} )

  • ( \sinh(x) = \frac{e^x - e^{-x}}{2} )

Derivatives of Hyperbolic Functions

  • ( \frac{d}{dx} \sinh(x) = \cosh(x) )

  • ( \frac{d}{dx} \cosh(x) = \sinh(x) )

Hyperbolic Identities

  • ( \cosh^2(x) - \sinh^2(x) = 1 )