Transcript 10.5: Base e and Natural Logarithms

10.5: Base e and Natural
Logarithms
-Definition of e & graph
-Evaluating e
-Definition of ln & graph
-Evaluating natural logs
-Equations with e and ln
-Compounding interest
-Inequalities
Definition of “e”
• Suppose I look at the following expression: (1 + (1/x))x
• On the calc, we can use the table feature to investigate what
happens for large values of x
• For large x, the expression seems to be approaching a value a
bit larger than 2.7… we call this value “e”, the natural base
• As x →∞, (1 + (1/x))x → e
• “e” is an irrational number, like pi
• “e” is often used in word problems involving growth or decay
that is “continuous”
• The graph of f(x) = ex represents exponential growth and the
y-intercept is at (0, 1) (recall e is approximately 2.71828
Use a calculator to evaluate
Keystrokes:
2nd
Answer: about 1.6487
[ex] 0.5
to four decimal places.
ENTER
1.648721271
Use a calculator to evaluate
Keystrokes:
2nd
Answer: about 0.0003
[ex] –8
to four decimal places.
ENTER
.0003354626
Use a calculator to evaluate each expression to four decimal
places.
a.
Answer: 1.3499
b.
Answer: 0.1353
The natural logarithm
• Recall from the last section that your calculator can
easily evaluate common logarithms (logs with base
10)
• Your calculator can also evaluate logarithms with a
base of e (ex. Loge30)
• The log with base e is called the natural logarithm,
and is written ln (LN)
• F(x) = ln x is the inverse of y = ex
• F(x) = ln x resembles a typical logarithmic graph; the
y-axis is an asymptote, the x-intercept is at (1,))
Use a calculator to evaluate In 3 to four decimal places.
Keystrokes:
LN
3
Answer: about 1.0986
ENTER
1.098612289
Use a calculator to evaluate In
Keystrokes:
LN
1÷4
Answer: about –1.3863
to four decimal places.
ENTER
–1.386294361
Use a calculator to evaluate each expression to four decimal
places.
a. In 2
Answer: 0.6931
b. In
Answer: –0.6931
Write an equivalent logarithmic equation for
Answer:
.
Write an equivalent exponential equation for
Answer:
Write an equivalent exponential or logarithmic equation.
a.
Answer:
b.
Answer:
Evaluate
Answer:
Evaluate
Answer:
.
Evaluate each expression.
a.
Answer: 7
b.
Answer:
Solving equations
• Similar to what we’ve done in 10.2 – 10.4, BUT
if you are taking a log of each side, use LN
rather than the common log to save yourself
one step (you can use the common log as
well.. Just takes 1 more step)
Solve
Original equation
Subtract 4 from each side.
Divide each side by 3.
Property of Equality
for Logarithms
Inverse Property of
Exponents and Logarithms
Divide each side by –2.
Use a calculator.
Answer: The solution is about –0.3466.
Check You can check this value by substituting –0.3466 into the
original equation or by finding the intersection of the graphs of
and
Solve
Answer: 0.8047
Interest
• Recall that earlier we saw an example
involving interest that was compounded
periodically (e.g., monthly, daily, etc.
• A(t) = P(1 + (r/n))nt
• Find the balance after 6 years if you deposit
$1800 in an account paying 3% interest that is
compounded monthly
• A(6) = 1800(1 + (.03/12))12*6
• A(6) = $2154.51
More on interest
• What about if the interest is compounded not
monthly,daily, or even every second, but
CONSTANTLY?
• We call this continuous compounding.. At ANY time
you can instantly calculate your new balance
• The formula we use for continuously compounding
interest is:
• A(t) = Pert
• This expression stems from the fact that:
• As x →∞, (1 + (1/x))x → e
Savings Suppose you deposit $700 into an account paying 6%
annual interest, compounded continuously.
What is the balance after 8 years?
Continuous compounding formula
Replace P with 700,
r with 0.06, and t with 8.
Simplify.
Use a calculator.
Answer: The balance after 8 years would be $1131.25.
How long will it take for the balance in your account to reach at
least $2000?
The balance is at least $2000.
A

2000
Replace A with 700e(0.06)t.
Write an inequality.
Divide each side by 700.
Property of Inequality
for Logarithms
Inverse Property of
Exponents and
Logarithms
Divide each side by 0.06.
Use a calculator.
Answer: It will take at least 17.5 years for the balance to
reach $2000.
Savings Suppose you deposit $700 into an account paying 6%
annual interest, compounded continuously.
a. What is the balance after 7 years?
Answer: $1065.37
b. How long will it take for the balance in your account to
reach at least $2500?
Answer: at least 21.22 years
Solve
Original equation
Write each side using
exponents and base e.
Inverse Property of
Exponents and Logarithms
Divide each side by 3.
Use a calculator.
Answer: The solution is 0.5496. Check this
solution using substitution or graphing.
Inequalities
• Again, similar to what we saw in 10.1 – 10.3
• Remember that for a log inequality, the
expression you are taking the log OF must be
positive
• Ex. Ln (x + 3) < 4
• X must be greater than -3
Solve
Original inequality
Write each side using
exponents and base e.
Inverse Property of
Exponents and Logarithms
Add 3 to each.
Divide each side by 2.
Use a calculator.
Answer: The solution is all numbers less than
7.5912 and greater than 1.5. Check
this solution using substitution.
Solve each equation or inequality.
a.
Answer: about 1.0069
b.
Answer: