Study Guide for Chapter 5: Inverse, Exponential, and
Logarithmic Functions
5.1
5.2
5.3
5.4
5.5
Inverse Functions
•
Only one-to-one functions have inverse functions
•
Horizontal line test for one-to-one functions
•
Theorem on Inverse Functions: g(f(x)) = x and f(g(y)) = y
•
Notation: f –1(x) => inverse function of f(x). Note: f –1(x) is not the same as f(x)-1 !
•
Domain and range of f and f -1
•
Graph of f(x) and f –1(x) are reflections of each other through the line y = x
Exponential Functions
•
Extending domain of exponential to any real number requires irrational numbers
•
Exponential functions are one-to-one
•
Sketching graphs of exponential functions: horizontal and vertical shifts, etc.
•
Exponential growth and exponential decay
The Natural Exponential Function
•
Compound interest, interest compounded continuously, & the natural exponential function
•
The number e = 2.71828… & the natural exponential function f(x) = ex
•
Law of Growth (or Decay) Formula: q(t) = q0ert
Logarithmic Functions
•
Logarithm of x with base a is defined by y = logax if & only if ay = x (for x>0)
•
Logarithmic functions are one-to-one
•
Sketching, shifting & reflecting graphs of logarithmic functions
•
Usual bases are 10 (common logarithms) and e (natural logarithms)
•
Use of log with base not given => common log; ln => natural log
Properties of Logarithms
•
Three Laws of Logarithms, corresponding to Laws of Exponents (see below)
•
logax is defined only for a>0 and a≠1, and x>0
•
Solving logarithmic functions; more on sketching & shifting their graphs
1
5.6
Exponential and Logarithmic Equations
•
To solve equations with variables in exponents, take log of both sides
•
…but valid only if all exponents have the same basis
•
Theorem on changing base: log b u =
log a u
(u>0; a, b>0 and a, b ≠ 1)
log a b
€
Logarithms vs. Exponents
In the tables below, a is a positive real number other than 1; u and w are any positive real
numbers; and x, z, and c are any real numbers.
Laws of Exponents vs. Laws of Logarithms
Exponents
1
a x+z = a x ⋅ a z
2
€
Logarithms
a x−z =
3
ax
az
a cx = (a x )c
€
log a u + log a w = log a uw
log a u − log a w = log a
€
u
w
log a u c = c log a u
€
Properties €
of Logarithms vs. Exponents
€
€
Exponents
Logarithms
1
a0 = 1
log a 1 = 0
2
a1 = a
log a a = 1
3
a log a u€= u
log a a x = x
€
€
€
€
Examples


2 log 2 7 = 7 
€
€
€
DAB, rev. April 2011
2
log10 10 = 1
log 2 2 7 = 7