Properties of Logarithms and Common Logarithms Sec 10.3 & 10.4 pg

Hallford © 2007Glencoe © 2003 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.

Hallford © 2007Glencoe © 2003 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 = log 3 (9)(27) = log log 3 27 log = = 5

Hallford © 2007Glencoe © 2003 Work this problem Use log 4 7 ≈ to evaluate log We can write log 4 28 as log 4 (7)(4). We then can say log 4 28 = log log 4 4. What is log 4 4? Remember it is 1! So log log 4 4 = ≈ 2.404

Hallford © 2007Glencoe © 2003 Another way to work these types of problems Using log 3 5 ≈ to evaluate log Let’s factor 135! 135 = 5 * = 5 * 3 3. log = log 3 5 * 3 3 log 3 5 * 3 3 = log log Remember log b b x = x so log = 3 So now log log ≈ ≈ 4.465

Hallford © 2007Glencoe © 2003 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 ≈ and log 4 30 ≈ to evaluate log 4 5 = log 4 30 – log 4 6≈ – ≈ 1.161

Hallford © 2007Glencoe © 2003 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 ≈ to evaluate log log = log log = 7 log 9 2≈ 7 (0.315) ≈ 2.205

Hallford © 2007Glencoe © 2003 Work these problems 2 log 10 6 – 1 / 3 log = log 10 x log – log 10 3 = log 10 x log = log 10 x 12 = x

Hallford © 2007Glencoe © 2003 Work this problem log 7 24 – log 7 (y + 5) = log = 8(y + 5) 24 = 8y = 8y -2 = y Let’s check! ☺

Hallford © 2007Glencoe © 2003 Work this problem 48 = 4p 12 = p

Hallford © 2007Glencoe © 2003 End of 10.3

Hallford © 2007Glencoe © 2003 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 = =

Hallford © 2007Glencoe © 2003 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/ = 2.196

Hallford © 2007Glencoe © 2003 Solve 7 p + 2 < p log 7 p + 2 < log 13 5 – p (p + 2) log 7 < (5 – p) log (p + 2) < 1.114(5 – p) 0.845p < 5.57 – 1.114p 1.959p < 3.88 p < 1.98 Let’s check for p = 1.9 log < log < ☺

Hallford © 2007Glencoe © 2003 Change of Base Formula Let’s find log 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

Hallford © 2007Glencoe © 2003 Evaluate log Now check! = The difference is due to rounding