The Exponential and Logarithmic Functions. Natural. Section 6.3. Euler’s Number.
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The Exponential and Logarithmic Functions
This value is so important in mathematics thatit has been given its own symbol, e, sometimes called Euler’s number. The number e hasmany of the same characteristics as π. Its decimal expansion never terminates or repeatsin a pattern. It is an irrational number.
Value of e
To eleven decimal places,
e = 2.71828182846
e = 2.718281828…
is an irrational number called the natural base.
Use your calculator to evaluate the following.
Round our answers to 4 decimal places.
The function f, represented by
f(x) = cex
is the natural exponential function, where c is the constant, and x is the exponent.
f(x) = cex
Properties of an natural exponential function:
Example 3 – Page 514
State the transformation of each function, horizontal asymptote, y-intercept, and domain and range for each function.
The function of the form P(t) = P0ektModels exponential growth if k > 0 and exponential decay when k < 0.
T = time
P0 = 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
Example 4 – Page 516
For each natural exponential function, identify the initial value, the continuous growth or decay rate, and the growth or decay factor.
What is the Natural Logarithmic Function?
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
Example 7 – Page 518
Evaluate the following.
The graph of y= lnxis a reflection of the graph of
y=exacross the liney= x.
f(x) = ln x
Example 8 – Page 519
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 2
V.A. x = 4
Domain: (4, ∞)
Range: (-∞, ∞)
Reflect x axis down 2
V.A. x = 0
Domain: (0, ∞)
Range: (-∞, ∞)
Example 9 – Page 521
Find the domain of each function algebraically.
f(x) = ln (x-31)
f(x) = ln (5.4 - 2x) + 3.2
(31, ∞ )
(-∞, 2.7 )