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: