Overview
This lecture introduces logarithms as the inverses of exponential functions, explains their properties, defines their notation, and covers how to determine their domains and convert between exponential and logarithmic forms.
Exponential Functions Recapv
- Exponential functions have the form ( y = a^x ) where ( a > 0 ), ( a \neq 1 ).
- They always pass through the point (0, 1), and (1, a).
- If ( a > 1 ), the graph increases; if ( 0 < a < 1 ), it decreases.
- Domain: all real numbers (( -\infty, \infty )); Range: ( (0, \infty) ).
- Horizontal asymptote at ( y = 0 ); outputs never negative or zero.
Inverses and the Origin of Logarithms
- Exponential functions are one-to-one and thus have an inverse.
- The inverse is found by swapping x and y; graphically, itβs a reflection over ( y = x ).
- The inverse of an exponential is called a logarithm.
- Key points swap: (0,1) becomes (1,0), (1,a) becomes (a,1).
- The vertical asymptote for logarithms is at ( x = 0 ).
Logarithms: Definition and Properties
- Logarithms answer: βTo what exponent must the base be raised to yield the argument?β
- General form: ( y = \log_a{x} ) means ( a^y = x ).
- Domain: ( x > 0 ); Range: all real numbers.
- You cannot have zero or negative arguments inside a logarithm.
- Key points: always pass through (1, 0) and (a, 1).
- Output can be negative, but argument must stay positive.
Converting Between Exponential and Logarithmic Forms
- ( a^b = c )βββ( \log_a{c} = b ).
- Logarithms separate base and exponent; exponentials combine them.
Special Logarithms: Common and Natural Logs
- Common logarithm: base 10, written ( \log{x} ) (base omitted).
- Natural logarithm: base ( e ), written ( \ln{x} ).
Solving and Composing Logarithms and Exponentials
- Logarithms and exponentials are inverses and βundoβ each other.
- Composing them (with the same base) gives the original input (e.g., ( \log_a(a^x) = x )).
- Evaluating log expressions often involves converting to exponentials or using properties.
- Every logarithm of 1 equals 0: ( \log_a{1} = 0 ).
Finding Domains of Logarithms
- The argument (input) of any logarithm must be strictly positive (( > 0 )).
- For transformed arguments, set the inside ( > 0 ) and solve for x.
- For rational arguments, use sign analysis to determine intervals where the argument is positive.
- Shifts and reflections in the argument affect the domain accordingly.
Key Terms & Definitions
- Exponential Function β A function of the form ( y = a^x ) where ( a > 0 ), ( a \neq 1 ).
- Logarithm (( \log_a{x} )) β The inverse function of an exponential, finds the exponent for a given base and argument.
- Argument β The input value inside the logarithm, must be positive.
- Base β The constant in exponential and logarithmic functions, must be positive and not equal to 1.
- Common Logarithm (( \log{x} )) β Logarithm with base 10.
- Natural Logarithm (( \ln{x} )) β Logarithm with base ( e ) (Eulerβs number, β 2.718).
- Domain β The set of allowable input values (for logarithms, where the argument is positive).
Action Items / Next Steps
- Practice converting between exponential and logarithmic forms.
- Find the domain for various logarithmic functions.
- Review special logs (( \log ), ( \ln )), their notation, and calculator usage.
- Prepare for graphing logarithmic functions and applying transformations in the next lesson.