Apply Properties of logarithms Lesson 4.5 Honors Algebra 2 Apply Properties of logarithms Lesson 4.5

Goals Goal Rubric Use properties of logarithms. Expand and condense a logarithmic expression. Use the change of base formula. Use properties of logarithms in real life. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Base

Properties of logarithms Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents.

Properties of logarithms Remember that to multiply powers with the same base, you add exponents. Logarithm of a product is equal to the sum of the logarithms.

Properties of logarithms Using the product property as previously shown is called expanding the logarithm. log 𝑎𝑏𝑐 = log 𝑎 + log 𝑏 + log 𝑐 The property can also be used in reverse to write a sum of logarithms as a single logarithm, called condensing the logarithm. log 𝑎 + log 𝑏 + log 𝑐 = log 𝑎𝑏𝑐

Properties of logarithms Remember that to divide powers with the same base, you subtract exponents. Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

Properties of logarithms Logarithm of a quotient is equal to the difference of the logarithms. The property above can also be used in reverse.

Properties of logarithms Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified. Caution

Properties of logarithms Because you can multiply logarithms, you can also take powers of logarithms. Logarithm of a power is the product of the power and the logarithm.

Summary Properties of Logarithms Product Property: Logarithm of a product is equal to the sum of the logarithms. Quotient Property: Logarithm of a quotient is equal to the difference of the logarithms. Power Property: Logarithm of a power is the product of the power and the logarithm.

The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. This is because the properties convert more complicated products, quotients, and exponential forms into simpler sums, differences, and products. This is called expanding a logarithmic expression. The procedure above can be reversed to produce a single logarithmic expression. This is called condensing a logarithmic expression.

Use properties of logarithms EXAMPLE 1 Use properties of logarithms 3 4 log Use 0.792 and 7 1.404 to evaluate the logarithm. a. 4 log 3 7 = 3 – 4 log 7 Quotient property 0.792 1.404 – Use the given values of 3 4 log 7. and = –0.612 Simplify. b. 4 log 21 = 4 log (3 7) Write 21 as 3 7. = 3 4 log + 7 Product property 0.792 1.404 + Use the given values of 3 4 log 7. and = 2.196 Simplify.

Use properties of logarithms EXAMPLE 1 Use properties of logarithms 3 4 log Use 0.792 and 7 1.404 to evaluate the logarithm. c. 4 log 49 72 = 4 log Write 49 as 72 4 log = 2 7 Power property 2(1.404) Use the given value of 7. 4 log = 2.808 Simplify.

Your Turn: for Example 1 5 6 log Use 0.898 and 8 1.161 to evaluate the logarithm. 1. 5 8 6 log –0.263 ANSWER 2. 6 log 40 2.059 ANSWER

Your Turn: for Example 1 5 6 log Use 0.898 and 8 1.161 to evaluate the logarithm. 6 log 3. 64 2.322 ANSWER 4. 6 log 125 2.694 ANSWER

EXAMPLE 2 Expand a logarithmic expression Expand 6 log 5x3 y SOLUTION = 5x3 y 6 log – Quotient property = 5 6 log x3 y – + Product property = 5 6 log x y – + 3 Power property

Condense a logarithmic expression EXAMPLE 3 Condense a logarithmic expression SOLUTION – log 9 + 3log2 log 3 = – log 9 + log 23 log 3 Power property = log (9 ) 23 – log 3 Product property = log 9 23 3 Quotient property = 24 log Simplify. The correct answer is D. ANSWER

Your Turn: for Examples 2 and 3 Expand 5. log 3 x4 . log 3 + 4 log x ANSWER

Your Turn: for Examples 2 and 3 Condense ln 4 + 3 ln 3 – ln 12. 6. ln 9 ANSWER

Change of Base Formula Most calculators calculate logarithms only in base 10 or base e. You can change a logarithm in one base to a logarithm in another base with the following formula.

Change of Base Formula Change to base 10: log 𝑏 𝑥 = log 𝑥 log 𝑏 Change to base e : log 𝑏 𝑥 = ln 𝑥 ln 𝑏

EXAMPLE 4 Use the change-of-base formula 3 log 8 Evaluate using common logarithms and natural logarithms. SOLUTION Using common logarithms: = log 8 log 3 0.9031 0.4771 3 log 8 1.893 Using natural logarithms: = ln 8 ln 3 2.0794 1.0986 3 log 8 1.893

Your Turn: for Example 4 Use the change-of-base formula to evaluate the logarithm. 5 log 8 7. about 1.292 ANSWER 8 log 14 8. about 1.269 ANSWER

Your Turn: for Example 4 Use the change-of-base formula to evaluate the logarithm. 26 log 9 9. about 0.674 ANSWER 10. 12 log 30 about 1.369 ANSWER

EXAMPLE 5 Use properties of logarithms in real life Sound Intensity 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 = log L(I) 10 I 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

Use properties of logarithms in real life EXAMPLE 5 Use properties of logarithms in real life SOLUTION Let I be the original intensity, so that 2I is the doubled intensity. Increase in loudness = L(2I) – L(I) Write an expression. = log 10 I 2I – Substitute. = 10 log – I 2I Distributive property = 2 10 log I – + Product property 10 log 2 = Simplify. 3.01 Use a calculator. ANSWER The loudness increases by about 3 decibels.

Your Turn: for Example 5 WHAT IF? In Example 5, suppose the artist turns up the volume so that the sound’s intensity triples. By how many decibels does the loudness increase? 11. ANSWER The loudness increases by about 4.771 decibels.