Chapter 5: Exponential and Logarithmic Functions 5.5: Properties and Laws of Logarithms

1. Chapter 5: Exponential and Logarithmic Functions5.5: Properties and Laws of Logarithms Essential Question: What are the three properties that simplify logarithmic expressions? Describe how to use them.

2. 5.5: Properties and Laws of Logarithms • Basic Properties of Logarithms • Logarithms are only defined for positive real numbers • Not possible for 10 or e to be taken to an exponent and result in a negative number • Log 1 = 0 and ln 1 = 0 • 100 = 1 & e0 = 1 • Log 10k = k and lnek = k • log10104 = k • 10k = 104 • k = 4 • 10log v = v and eln v = v • 10log 22 = v • log10v = log 22 • v = 22

3. 5.5: Properties and Laws of Logarithms • Solving Equations by Using Properties of Logarithms • ln(x + 1) = 2 • Method #1 • e2 = x + 1 • e2 – 1 = x • x ≈ 6.3891 • Method #2 • eln(x + 1) = e2 • x + 1 = e2 • See method #1 above

4. 5.5: Properties and Laws of Logarithms • Product Law of Logarithms • Law of exponents states bmbn = bm+n • Because logarithms are exponents: • log (vw) = log v + log w • ln (vw) = ln v + ln w • Proof: • vw = 10log v • 10log w = 10log v + log w • vw = 10log vw • Taking from above: • 10log v + log w = 10log vw • log v + log w = log vw • Proof of ln/e works the same way

5. 5.5: Properties and Laws of Logarithms • Product Law of Logarithms (Application) • Given that log 3 = 0.4771 and log 11 = 1.0414find log 33 • log 33 = log (3 • 11) = log 3 + log 11 = 0.4771 + 1.0414 = 1.5185 • Given that ln 7 = 1.9459 and ln 9 = 2.1972find ln 63 • ln 63 = ln (7 • 9) = ln 7 + ln 9 = 1.9459 + 2.1972 = 4.1431

6. 5.5: Properties and Laws of Logarithms • Quotient Law of Logarithms • Law of exponents states • Because logarithms are exponents: • log ( ) = log v – log w • ln ( ) = ln v – ln w • Proof is the same as the Product Law

7. 5.5: Properties and Laws of Logarithms • Quotient Law of Logarithms (Application) • Given that log 28 = 1.4472 and log 7 = 0.8451find log 4 • log 4 = log (28 / 7) = log 28 – log 7 = 1.4472 – 0.8451 = 0.6021 • Given that ln 18 = 2.8904 and ln 6 = 1.7918find ln 3 • ln 3 = ln (18 / 6) = ln 18 – ln 6 = 2.8904 – 1.7918 = 1.0986

8. 5.5: Properties and Laws of Logarithms • Power Law of Logarithms • Law of exponents states (bm)k = bmk • Because logarithms are exponents: • log (vk) = k log v • ln (vk) = k ln v • Proof: • v = 10log v→ vk = (10log v)k = 10k log v • vk = 10log vk • Taking from above: • 10k log v = 10log vk • k log v = log vk • Proof of ln/e works the same way

9. 5.5: Properties and Laws of Logarithms • Power Law of Logarithms (Application) • Given that log 6 = 0.7782 find log • log = log 6½ = ½ log 6 = ½ (0.7782) = 0.3891 • Given that ln 50 = 3.9120 find ln

10. 5.5: Properties and Laws of Logarithms • Simplifying Expressions • Write as a single logarithm: • ln 3x + 4 ln x – ln 3xy

11. 5.5: Properties and Laws of Logarithms • Simplifying Expressions • Write as a single logarithm:

12. 5.5: Properties and Laws of Logarithms • Assignment • Page 369 • Problems 1-25, odd problems • Show work