"The greater part of our happiness or misery depends on our dispositions, and not on our circumstances." Martha Dandridge Custis Washington 1731 – 1802

Chapter 3 Exponential and Logarithmic Functions

Day III Properties of Logarithms (3.3)

Logarithmic functions are often used to model scientific observations like human memory.

GOAL I. To rewrite logarithmic functions with a different base

I. Change of Base

Calculators only have two types of log keys. The common log and the natural log. The bases are 10 and e, respectively.

Change-of-Base Formula

Let a, b, and x be positive real numbers such that a  1 and b  1. Then log a x can be converted to a different base as follows:

Base b log a x = log b x log b a

Base 10 log a x = log x log a

Base e log a x = ln x ln a

Example 1. Changing Bases Using Common Logarithms

Using a calculator and the common log setting, evaluate the expression to 1/ log 7 4 = log 4/log 7 =

Your Turn

Using a calculator and the common log setting, evaluate the expression to 1/ log 1/4 5 = 2. log =

Example 2. Changing Bases Using Natural Logarithms

Using a calculator and the natural log setting, evaluate the expression to 1/ log 7 4 = ln 4/ln 7 =

Your Turn

Using a calculator and the natural log setting, evaluate the expression to 1/ log 1/4 5 = 2. log =

GOAL II. To use properties of logarithms to evaluate or rewrite logarithmic expressions

II. Properties of Logarithms

Summative Math Algebra 2 Standard Students understand the properties of logarithms (log laws).

Let a be a positive number such that a  1, and let n be a real number. If u and v are positive real numbers, the following properties are true.

1. log a (uv) = log a u + log a v log (uv) = log u + log v ln (uv) = ln u + ln v

2. log a = log a u – log a v log = log u – log v ln = ln u – ln v uvuv uvuv uvuv

3. log a u n = n log a u log u n = n log u ln u n = n ln u

Example 3. Using Properties of Logarithms

Use the properties of logarithms and the given values to find the logarithm indicated. NO CALCULATORS!!! log 7  0.8 log 8  0.9 log 12  1.1

1.log = 7878 log 7 – log 8  0.8 – 0.9  – 0.1

2.log 64 = log 8 2  2 log 8  2(0.9)  1.8

3.log 96 = = log 8 + log 12   2.0 log 8  12

Your Turn

1.log = 7 12 log 7 – log 12  0.8 – 1.1  – 0.3

2.log 49 = log 7 2  2 log 7  2(0.8)  1.6

3.log 1008 = log 7 + log 12 2  (1.1)  3.0 = log 7 + 2log 12 

Example 4. Using Properties of Logarithms

Use the properties of logarithms and the given values to find the logarithm indicated. NO CALCULATORS!!!

1.log 9 = log = -2B = -2log log 9 7 = A log 9 4 = B log 9 10 = C

2.log = =log log = S + 2 = log log 7 7 log 7 6 = R log 7 8 = S log 7 10 = T log 8 8  7 2 = S + 2(1)

3.log 8 = log 8 = log log 8 8 – log = log log 8 12 = P log 8 5 = Q log 8 9 = R  = P + 1 – 2log 8 9 = P + 1 – 2R

Your Turn

1.log 5 = log = -R = -1log log 5 12 = R log 5 9 = S log 5 11 = T

2.log = =3log 8 9 = 3B log 8 6 = A log 8 9 = B log 8 10 = C log 8 9 3

3.log 7 = log 7 = log log 7 10 – log = log log 7 3 = X log 7 8 = Y log 7 10 = Z  = X + Z – 2log 7 8 = X + Z – 2Y

What do you get when you cross a fawn and a hornet? Bambee

GOAL III. To use properties of logarithms to expand or condensed logarithmic expressions

III. Rewriting Logarithmic Expressions

Summative Math Algebra 2 Standard Students use the properties of logarithms to identify their approximate values (expanding).

Example 5. Expanding Logarithmic Expressions

Use properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. Assume all variables are positive.

1.ln, x > 1 x 2 – 1 x 3 = ln (x + 1)(x – 1) x 3 = ln(x + 1) + ln (x – 1) – ln x 3 = ln(x + 1) + ln (x – 1) – 3ln x

2.ln  x 2 (x + 2) = ln x 2 (x + 2) = ½ [ln x 2 + ln (x + 2)] = ½ [2ln x + ln (x + 2)] ½ = ln x + ½ ln (x + 2)

Your Turn

Use properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. Assume all variables are positive.

1.ln x  x = ln x (x 2 + 1) 1/2 = ln x – ln (x 2 + 1) ½ = ln x – ½ln (x 2 + 1)

2.ln x2y3x2y3 = ln x2x3x2x3 = ½[ln x 2 – ln y 3 ] = ½ [2ln x – 3ln y] ½ = ln x – ln y 3232

Summative Math Algebra 2 Standard Students use the properties of logarithms to simplify logarithmic numeric expressions (condensing).

Example 6. Condensing Logarithmic Expressions

Condense the expression to the logarithmic of a single quantity.

1. 4[lnz + ln(z + 2)] – 2ln(z – 5) = ln [z(z + 2)] 4 (z – 5) 2 = ln z 4 (z + 2) 4 (z – 5) 2

Your Turn

1.2ln 8 + 5ln z = ln ln z 5 = ln 64z 5

2. 2[lnx – ln(x + 1) – ln(x – 1)] = ln x 2 (x + 1)(x – 1) = ln x 2 x 2 – 1 = 2 lnx – [ln(x + 1) + ln(x – 1)]

GOAL IV. To use logarithmic functions to model and solve real-life applications

IV. Applications

Logarithmic functions are often used to model scientific observations like human memory.

Example 7. Finding a Mathematical Model

Students participating in a psychological experiment attended several lectures and were given an exam.

Every month for a year after the exam, the students were retested to see how much of the material they remembered.

The average score of the group can be modeled by the memory model f(t) = 90 – 15 log (t + 1), 0  t  12 where t is the time in months.

1. What was the average score on the original exam (t = 0)? f(0) = 90 – 15 log (t + 1) 0 f(0) = 90 – 15 log points

2. What was the average score after six months? f(t) = 90 – 15 log (t + 1) 6 6 f(6) = 90 – 15 log 7 f(6)  77 points

Your Turn

3. What was the average score after 12 months? f(12) = 90 – 15 log (12 + 1) f(12) = 90 – 15 log 13 f(12)  73 points

4. When will the average score decrease to 75? f(t) = 90 – 15 log (t + 1) = – 15 log (t + 1) 1 = log (t + 1) 10 1 = t months

Acupuncture is a jab well done. Pun for the Day