Exponential Functions

The ex pression a ^r for a rational number r = m/n has been discussed and is
defined as:

What about a^x for a ir rational number x?

• it can be approximated by using rational approximation of x

• the Laws of Exponents apply to a^x for any real number x, and other
listed properties

• Exp onential function

The equation f (x) = b^x, b > 0, b ≠ 1 defines an exponential function for
each different constant b, called the base.

The domain of f is the set of all real numbers. The range of f is the set of
all positive real numbers.

• Basic Properties of Exponential Functions

1. All graphs pass through the point (0,1).
2. All graphs are continuous with no holes or jumps.
3. The x-axis is a horizontal asymptote.
4. Increasing function for b > 1.
5. Decreasing function for 0 < b < 1.
6. All exponential functions are one-to-one.

•Transformation Techniques for Graphing Exponential Functions

 • vertical shift - f(x) = b^x+c
 • horizontal shift - f(x) = b^x+c
 • vertical and horizontal scaling - f(x) = cb^x and f(x) = b^cx

• Exponential Functions are Common Model in Many Applications

• Increasing exponential functions (b > 1) are called growth models and are
used for applications like population growth, death rate, and compound
interest.

– Doubling Time Growth Model:

– Compound Interest Formula :

• Decreasing exponential functions (0 < b < 1) are called decay models
and are used for applications like radioactive decay, survival rate, atmospheric
pressure, and failure time of electronics equipment.

– Half-Life Decay Model:

• Exponential Function with Base e

Many common problems in nature require the use of an exponential function
with base e,

where e is a very important irrational number in mathematics (comparable to
π) named after Euler defined as

Note: The exponential function with base e is often referred to as the exponential
function.

• Common Applications

• Most exponential growth and decay problems are modeled with the exponential
function.

– Carbon Dating:

– Continuous Compound Interest /Population Growth:

• Limited growth problems (learning skills, company growth, etc.)
• Logistic growth problems (long- term population growth, epidemics, sales
of new products, etc.)

Logarithmic Functions

• The logarithmic function with base b is the inverse of the exponential
function with base b. It is defined (for b > 0, b ≠ 1) as f(x) = log_bx, where
y = log_bx is equivalent to x = b^y.

Note: log_bx is the exponent to which b must be raised to obtain x.

• f(x) = log_bx is the inverse of b^x

– the domain of f is equal to the range of b^x (set of all positive reals)
– the range of f is equal to the domain of bx^x (set of all reals)
– the graph of f is a reflection about y = x of the graph of y = b^x

• Basic Properties of Logarithmic Functions

1. All graphs pass through the point (1,0).
2. All graphs are continuous with no holes or jumps.
3. The y-axis is a vertical asymptote.
4. Increasing for b > 1. Decreasing for 0 < b < 1.
5. All logarithmic functions are one-to-one.
6. Miscellaneous Properties:

• Transformation Techniques for Graphing Logarithmic Functions

vertical shift - f(x) = log_b(x) + c
• horizontal shift - f(x) = log_b(x + c)
• vertical and horizontal scaling - f(x) = c log_b(x) and f(x) = log_b(cx)

• Common and Natural Logarithms

The common (base 10) and natural (base e) logarithms are used almost exclusively
now. They are denoted as:

• Common Logarithm: log x = log₁₀x.
• Natural Logarithm: ln x = log_ex.

Any logarithm of base b can be expressed in terms of the common or natural
logarithm, or any other base.

• Common Applications

• The logarithmic function historically has been used to reduce the computational
strain in research involving outcomes with an extremely large
scale of values . Modern calculators and computers have solved this problem
in many (but not all) cases.

• The logarithmic function is still commonly used to rescale data.

– Decibel Scale:

– Richter Scale:

– Rocket Velocity:

• Exponential and Logarithmic Equations
Equations involving exponential and logarithmic functions are called exponential
and logarithmic equations, respectively. They can often be solved using
the basic properties of exponential and logarithmic functions.