section 6.3. this value is so important in mathematics that it has been given its own symbol, e,...
TRANSCRIPT
The Exponential and Logarithmic Functions
Section 6.3
Natural
Euler’s NumberThis value is so important in mathematics that it has been given its own symbol, e, sometimes called Euler’s number. The number e has many of the same characteristics as π. Its decimal expansion never terminates or repeats in a pattern. It is an irrational number.
To eleven decimal places,
e = 2.71828182846
Value of e
The base e, which is approximately e = 2.718281828…
is an irrational number called the natural base.
Definition
Use your calculator to evaluate the following. Round our answers to 4 decimal places.
7.3891 0.1353
1.3499
103.0455
Example 1 – Page 511
The Natural Exponential Function
The function f, represented by
f(x) = cex
is the natural exponential function, where c is the constant, and x is the exponent.
Properties of Natural Exponential Function
Properties of an natural exponential function:
• Domain: (-∞, ∞)
• Range: (0, ∞)
• y-intercept is (0,c)
• f increases on (-∞, ∞)
• The negative x-axis is a horizontal asymptote.
• f is 1-1 (one-to-one) and therefore has an inverse.
Example
f(x) = cex
State the transformation of each function, horizontal asymptote, y-intercept, and domain and range for each function.
•1 unit right, down 3 units•h.a. y = -3•y-int: f(0) = -2.6•Domain: (-∞, ∞)•Range: (-3, ∞).
• reflect x-axis•h.a. y = 0•y-int: f(0) = -1•Domain: (-∞, ∞)•Range: (-∞, 0).
• reflect y-axis, down 5•h.a. y = -5•y-int: f(0) = -4•Domain: (-∞, ∞)•Range: (-5, ∞).
Example 3 – Page 514
Natural Exponential Growth and Decay
The function of the form P(t) = P0ekt Models exponential growth if k > 0 and exponential decay when k < 0.
T = timeP0 = the initial amount, or value of P at time 0, P > 0
k = is the continuous growth or decay rate (expressed as a decimal)
ek = growth or decay factor
For each natural exponential function, identify the initial value, the continuous growth or decay rate, and the growth or decay factor.
.
•Initial Value : 100
•Growth Rate: 2.5%
•Growth Factor: = 1.0253
•Initial Value : 500
•Decay Rate: -7.5%
•Decay Factor: = 0.9277
Example 4 – Page 516
Ricky bought a Jeep Wrangler in 2003. The value of his Jeep can by modeled by V(t)=25499e-0.155t where t is the number of years after 2003.
a) Find and interpret V(0) and V(2).
b) What is the Jeep’s value in 2007?
Example (Problem 57– Page 526
What is the Natural Logarithmic Function?What is the Natural Logarithmic Function?
• Logarithmic Functions with Base 10 are called “common logs.”
• log (x) means log10(x) - The Common Logarithmic Function
• Logarithmic Functions with Base e are called “natural logs.”
• ln (x) means loge(x) - The Natural Logarithmic Function
Let x > 0. The logarithmic function with base e is defined as y = logex. This function is called the natural logarithm and is denoted by y = ln x.
y = ln x if and only if x=ey.
Definition
Basic Properties of Natural Logarithms
ln (1)
ln (e)
ln (ex)
ln (1) = loge(1) = 0 since e0= 1
ln(e) = loge (e) = 1 since 1 is the
exponent that goes on e to produce e1.
ln (ex) = loge ex = x since ex= ex
= x
Evaluate the following.
Example 7 – Page 518
Graphs: Natural Exponential Function and Natural Logarithmic Function.The graph of y = lnx is a reflection of the graph of y =ex across the line y = x.
Properties of Natural Logarithmic Functions
• Domain: (0, ∞)
• Range: (-∞, ∞)
• x-intercept is (1,0)
• Vertical asymptote x = 0.
• f is 1-1 (one-to-one)
f(x) = ln x
For each function, state the transformations applied to y = lnx. Determine the vertical asymptote, and the domain and range for each function.
b. f(x) = ln(x-4) + 2 c. y = -lnx - 2
4 Right, Shift Up 2V.A. x = 4Domain: (4, ∞)Range: (-∞, ∞)
Reflect x axis down 2V.A. x = 0Domain: (0, ∞)Range: (-∞, ∞)
Example 8 – Page 519
Find the domain of each function algebraically.
(31, ∞ )
(-∞, 2.7 )
f(x) = ln (x-31)
f(x) = ln (5.4 - 2x) + 3.2
Example 9 – Page 521