Properties of Logarithms and Common Logarithms

1. Properties of Logarithms and Common Logarithms Sec 10.3 & 10.4 pg. 541 - 549

2. Objectives • TLWBAT • simplify and evaluate expressions using the properties of logarithms, and • solve logarithmic equations using the properties of logarithms. • solve exponential equations and inequalities using common logarithms, and • evaluate logarithmic expressions using the Change of Base formula.

3. Product Property of Logarithms • The logarithm of a product is the sum of the logarithms of its factors. • For all positive numbers m, n, and b, where b ≠ 1, log b mn = log b m + log b n • Example • log 3 243 = log 3 (9)(27) = log 3 9 + log 3 27 • log 3 243 = 2 + 3 = 5

4. Work this problem • Use log 4 7 ≈ 1.404 to evaluate log 4 28. • We can write log 4 28 as log 4 (7)(4). • We then can say log 4 28 = log 4 7 + log 4 4. • What is log 4 4? • Remember it is 1! • So log 4 7 + log 4 4 = 1.404 + 1 ≈ 2.404

5. Another way to work these types of problems • Using log 3 5 ≈ 1.465 to evaluate log 3 135. • Let’s factor 135! • 135 = 5 * 27 • 135 = 5 * 3 3. • log 3 135 = log 3 5 * 3 3 • log 3 5 * 3 3 = log 3 5 + log 3 3 3 • Remember log b b x = x so log 3 3 3 = 3 • So now log 3 5 + log 3 3 3 ≈ 1.465 + 3 ≈ 4.465

6. Quotient Property of Logarithms • The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. • For all positive numbers n, m, and b, where b ≠ 1, Use log 4 6 ≈ 1.292 and log 4 30 ≈ 2.453 to evaluate log 4 5 = log 4 30 – log 4 6 ≈ 2.453 – 1.292 ≈ 1.161

7. Power Property of Logarithms • The logarithm of a power is the product of the logarithm and the exponent. • For any real number p and positive numbers m and b, where b ≠ 1, • log b m p = p log b m Use log 9 2 ≈ 0.315 to evaluate log 9 128 log 9 128 = log 9 2 7 log 9 128 = 7 log 9 2 ≈ 7 (0.315) ≈ 2.205

8. Work these problems • 2 log 10 6 – 1/3 log 10 27 = log 10 x • log 10 36 – log 10 3 = log 10 x log 10 12 = log 10 x 12 = x

9. Work this problem • log 7 24 – log 7 (y + 5) = log 7 8 Let’s check! 24 = 8(y + 5) ☺ 24 = 8y + 40 -16 = 8y -2 = y

10. Work this problem 48 = 4p 12 = p

11. End of 10.3

12. Common Logarithms • Logarithms to the base 10 are called common logarithms. • You can calculate base 10 logarithms using your calculator. • Find log 23 • log 23 = 1.36 • log x = 2.3. Find x • If log x = 2.3, x = 10 2.3 = 199.53

13. Solve Problems • Solve 4 x = 21 • Take the log of both sides • log 4 x = log 21. • Use our power property • x log 4 = log 21. • Now divide both sides by log 4 • x = log 21/log 4 = 1.322/0.6021 = 2.196

14. Solve 7 p + 2< 13 5 - p • log 7 p + 2< log 13 5 – p • (p + 2) log 7 < (5 – p) log 13 • 0.845(p + 2) < 1.114(5 – p) • 0.845p + 1.69 < 5.57 – 1.114p • 1.959p < 3.88 • p < 1.98 • Let’s check for p = 1.9 • log 7 3.9 < log 13 3.1 • 3.296 < 3.453 ☺

15. Change of Base Formula • Let’s find log 7 33. • From our definition of logs we should know that this will be somewhere between 1 and 2, because 7 1 = 7 and 7 2 = 49. • We can use our change of base formula to use common logs to evaluate this expression. • Change of base formula We normally use 10 as our b since we can use the calculator to calculate common logs. = 1.516/0.845 = 1.79

16. Evaluate log 11 2435 3.252 Now check! The difference is due to rounding 11 3.252 = 2435.617