7.7 – Base e and Natural Logarithms

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Presentation transcript:

7.7 – Base e and Natural Logarithms

The number e. The history of mathematics is marked by the discovery of the special numbers such as zero, π and i. Another number, like π and i, that is denoted by a letter is the number e.

Natural Logarithm Natural Logarithm: log with base e: loge x Notation: ln x e is the base e is the base x is the argument y is the exponent

Example 1: Write in logarithmic form. A. ey = 23 B. e4 = x

Example 2: Write in exponential form. A. ln 25 = y B. ln x = 47

Example 3a: Solve Base e Equations Round to the nearest thousandth. 3e–2x + 4 = 10.

Example 3b: Solve the exponential equation. Round to the nearest thousandth. 6e3x + 5 = 15

Example 4: Condensing A. Write 4 ln 3 + ln 6 as a single logarithm. B. Write 2 ln 3 + ln 4 - ln y as a single logarithm.

Example 5a: Solve Natural Log Equations Solve. Round to the nearest thousandth. ln 5x + 3 = 6

Example 5b: Solve Natural Log Equations Solve. Round to the nearest thousandth. 3 ln (6x – 2) = 12

Example 5c: Solve Natural Log Equations Solve. Round to three decimals. 3 ln 5 + 2 ln x = 6

Example 5d: Solve Natural Log Equations Solve. Round to three decimals. ln (x + 5) = ln (x – 1) – ln (x + 1)

Example 6a: Solve Natural Log Inequalities Solve. Round to the nearest thousandth. ln (4x – 2) < 7

Example 6b: Solve Natural Log Inequalities Solve. Round to the nearest thousandth. ln (3x + 1)2 > 8

Example 7: Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. What is the balance after 8 years?

Example 8: Suppose you deposit $700 into an account paying 2.4% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $1200?