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Properties of Logarithms Chapter 3 Sec 3 Properties of Logarithms

Change-of-base formula Essential Question How do you rewrite logarithmic expressions to simplify or evaluate them? Key Vocabulary: Change-of-base formula

Change of Base Formula This allows you to write equivalent logarithmic expressions that have different bases. For example change base 3 into base 10

Change of Base Example 1 Change bases using common logarithms. Then approximate its value. Change bases using natural logarithms. Then approximate its value.

Properties of Logarithms Let a be a positive number such that a ≠ 1, and let n be a real number. If u and v are positive real numbers, the following properties are true. Product Property: Quotient Property: Power Property:

Example 2 Write each logarithm in terms of ln 2 and ln 3. Use properties of logarithms to verify:

Example 3 Expand each logarithmic expression.

Example 4 Condense each logarithmic expression.

Essential Question How do you rewrite logarithmic expressions to simplify or evaluate them?

Solving Exponential and Logarithmic Equations Chapter 3 Sec 4 Solving Exponential and Logarithmic Equations Essential Question How do you solve exponential and logarithmic equations?

Solving Equations One-to-One ax = ay if and only if x = y loga x = loga y if and only if x = y Inverse Properties

2x = 32 ln x – ln 3 = 0 ex = 7 ln x = –3 log x = –1 2x = 25 Example 1 Original Equation Rewritten Equation Solution Property 2x = 32 ln x – ln 3 = 0 ex = 7 ln x = –3 log x = –1 2x = 25 ln x = ln 3 3–x = 32 ln ex = ln 7 eln x = e–3 10log x = 10–1 x = 5 x = 3 x = –2 x = 7 x = e–3 x = 10–1 = 0.1 One-to-One Inverse

Solve each equation. a. ex = 72 b. 3(2x) = 42 ln ex = ln 72 2x = 14 Example 2 a. ex = 72 ln ex = ln 72 x = ln 72 ~ 4.28 b. 3(2x) = 42 2x = 14 log22x = log214 x = log214

Solve the equation. Example 3

Solve the equation using quadratics. Example 4

Solve each equation. Example 5 a. ln 3x = 2 eln 3x = e2 3x = e2

Solve the equation check for extraneous solutions. Example 6 Solve the equation check for extraneous solutions. Check: X ln (–1) is invalid.

Solve each equation. Example 5 You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?

How do you solve exponential and logarithmic equations? Essential Question How do you solve exponential and logarithmic equations?

Chapter 3.3 -3.4 Text Book Daily Assignment Pgs 211 – 212 #1 – 45 and 59 – 73 Mode 4 (1,5,9,13…73) Pgs 221 – 222 #1 – 21 Mode 4; #29 – 49 Mode 4; #85 – 97 Mode 4 Read Section 3.5

Ch 3a Pop Quiz