4.5 Apply Properties of Logarithms p. 259 What are the three properties of logs? How do you expand a log? Why? How do you condense a log?

Properties of Logarithms

Use log 5 3≈.683 and log 5 7≈1.209 log 5 21 = log 5 (3·7)= log log 5 7≈ = 1.892

Use log 5 3≈.683 and log 5 7≈1.209 Approximate: log 5 49 = log = 2 log 5 7 ≈ 2(1.209)= 2.418

2. 6 log 40 = 6 log (8 5) = 8 6 log = Write 40 as 8 5. Product property Simplify. ≈

Expanding Logarithms You can use the properties to expand logarithms. log 2 = log 2 7x 3 - log 2 y = log log 2 x 3 – log 2 y = log ·log 2 x – log 2 y

Your turn! Expand: log 5mn = log 5 + log m + log n Expand: log 5 8x 3 = log ·log 5 x

Condensing Logarithms log log2 – log 3 = log 6 + log 2 2 – log 3 = log (6·2 2 ) – log 3 = log = log 8

SOLUTION Evaluate using common logarithms and natural logarithms. Using common logarithms: Using natural logarithms: 3 log 8 = log 8 log log 8 = ln 8 ln

What are the three properties of logs? Product—expanded add each, Quotient— expand subtract, Power—expanded goes in front of log. How do you expand a log? Why? Use “log b ” before each addition or subtraction change. Power property will bring down exponents so you can solve for variables. How do you condense a log? Change any addition to multiplication, subtraction to division and multiplication to power. Use one “log b ”

For a sound with intensity I (in watts per square meter), the loudness L(I) of the sound (in decibels) is given by the function = logL(I)L(I)10 I 0 I Sound Intensity 0 I where is the intensity of a barely audible sound (about watts per square meter). An artist in a recording studio turns up the volume of a track so that the sound’s intensity doubles. By how many decibels does the loudness increase? 10 –12

Product property Simplify. SOLUTION Let I be the original intensity, so that 2I is the doubled intensity. Increase in loudness = L(2I) – L(I) = log10 I 0 I log10 2I2I 0 I – I 0 I 2I2I 0 I = log – = 210log I 0 I – I 0 I + ANSWER The loudness increases by about 3 decibels. 10 log 2 = 3.01 Write an expression. Substitute. Distributive property Use a calculator.

4.5 Assignment page 262, 7-41 odd

Properties of Logarithms Day 2 What is the change of base formula? What is its purpose?

Your turn! Condense: log ·log 5 t = log 5 7t 3 Condense: 3log 2 x – (log log 2 y)= log 2

Change of base formula: a, b, and c are positive numbers with b≠1 and c≠1. Then: log c a = log c a = (base 10) log c a = (base e)

Examples: Use the change of base to evaluate: log 3 7 = (base 10) log 7 ≈ log (base e) ln 7 ≈ ln

Use the change-of-base formula to evaluate the logarithm. 5 log 8 SOLUTION 5 log 8 = log 8 log log 14 SOLUTION 8 log 14 = log 14 log

What is the change of base formula? What is its purpose? Lets you change on base other than 10 or e to common or natural log.

4.5 Assignment Day 2 Page 262, even, odd