Objective – TSW solve equations that contain “e” and natural logarithms.
10.5: Base e and Natural Logarithms
30
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
description
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 - PowerPoint PPT Presentation
Transcript of 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
Answer: about 1.6487
ENTER2ndKeystrokes: [ex] 0.5 1.648721271
Use a calculator to evaluate to four decimal places.
Answer: about 0.0003
ENTER2ndKeystrokes: [ex] –8 .0003354626
Use a calculator to evaluate to four decimal places.
Use a calculator to evaluate each expression to four decimal places.
a.
b.
Answer: 1.3499
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: ENTERLN 3 1.098612289
Answer: about 1.0986
Keystrokes: ENTERLN 1 ÷ 4 –1.386294361
Answer: about –1.3863
Use a calculator to evaluate In to four decimal places.
Use a calculator to evaluate each expression to four decimal places.
a. In 2
b. In
Answer: 0.6931
Answer: –0.6931
Answer:
Write an equivalent logarithmic equation for .
Answer:
Write an equivalent exponential equation for
Answer:
Answer:
Write an equivalent exponential or logarithmic equation.
a.
b.
Evaluate
Answer:
Evaluate .
Answer:
Evaluate each expression.
a.
b.
Answer: 7
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 equationSubtract 4 from each side.Divide each side by 3.Property of Equality for Logarithms
Divide each side by –2.
Use a calculator.Answer: The solution is about –0.3466.
Inverse Property of Exponents and Logarithms
Check You can check this value by substituting –0.3466 into the original equation or by finding the intersection of the graphs of
and
Answer: 0.8047
Solve
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?
Answer: The balance after 8 years would be $1131.25.
Continuous compounding formula
Replace P with 700, r with 0.06, and t with 8.
Simplify.
Use a calculator.
How long will it take for the balance in your account to reach at least $2000?
Divide each side by 700.
Property of Inequality for Logarithms
The balance is at least $2000.
Write an inequality.2000 Replace A with 700e(0.06)t.A
Inverse Property of Exponents and Logarithms
Answer: It will take at least 17.5 years for the balance to reach $2000.
Use a calculator.
Divide each side by 0.06.
Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously.
a. What is the balance after 7 years?
b. How long will it take for the balance in your account toreach at least $2500?
Answer: $1065.37
Answer: at least 21.22 years
Answer: The solution is 0.5496. Check this solution using substitution or graphing.
Original equation
Write each side using exponents and base e.
Inverse Property of Exponents and Logarithms
Use a calculator.
Divide each side by 3.
Solve
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.
Use a calculator.
Divide each side by 2.
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.
b.
Answer: about 1.0069
Answer:
