Properties of Logarithms Section 3.3

Objectives Rewrite logarithms with different bases. Use properties of logarithms to evaluate or rewrite logarithmic expressions. Use properties of logarithms to expand or condense logarithmic expressions.

History of Logarithms John Napier, a 16 th Century Scottish scholar, contributed a host of mathematical discoveries. John Napier (1550 – 1617)

He is credited with creating the first computing machine, logarithms and was the first to describe the systematic use of the decimal point. Other contributions include a mnemonic for formulas used in solving spherical triangles and two formulas known as Napier's analogies. “In computing tables, these large numbers may again be made still larger by placing a period after the number and adding ciphers.... In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many ciphers after it as there are figures after the period.”

Napier lived during a time when revolutionary astronomical discoveries were being made. Copernicus’ theory of the solar system was published in 1543, and soon astronomers were calculating planetary positions using his ideas. But 16 th century arithmetic was barely up to the task and Napier became interested in this problem. Nicolaus Copernicus ( )

Even the most basic astronomical arithmetic calculations are ponderous. Johannes Kepler ( ) filled nearly 1000 large pages with dense arithmetic while discovering his laws of planetary motion! Johannes Kepler ( ) A typical page from one of Kepler’s notebooks

Napier’s Bones In 1617, the last year of his life, Napier invented a tool called “Napier's Bones” which reduces the effort it takes to multiply numbers. “ Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions... I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.”

Logarithms Appear The first definition of the logarithm was constructed by Napier and popularized by a pamphlet published in 1614, two years before his death. His goal: reduce multiplication, division, and root extraction to simple addition and subtraction. Napier defined the "logarithm" L of a number N by: N==10 7 (1-10 (-7) ) L This is written as NapLog(N) = L or NL(N) = L While Napier's definition for logarithms is different from the modern one, it transforms multiplication and division into addition and subtraction in exactly the same way.

Logarithmic FAQs Logarithms are a mathematical tool originally invented to reduce arithmetic computations. Multiplication and division are reduced to simple addition and subtraction. Exponentiation and root operations are reduced to more simple exponent multiplication or division. Changing the base of numbers is simplified. Scientific and graphing calculators provide logarithm functions for base 10 (common) and base e (natural) logs. Both log types can be used for ordinary calculations.

Logarithmic Notation For logarithmic functions we use the notation : log a (x) or log a x This is read “log, base a, of x.” Thus, y = log a x means x = a y And so a logarithm is simply an exponent of some base.

Change-of-Base Formula Example: Approximate log Only logarithms with base 10 or base e can be found by using a calculator. Other bases require the use of the Change-of-Base Formula. Change-of-Base Formula If a  1, and b  1, and M are positive real numbers, then 10 is used for both bases.

The Change-of-Base Rule ProofLet Change-of-Base Rule For any positive real numbers x, a, and b, where a  1 and b  1,

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

Change-of-Base Formula Example: Approximate the following logarithms.

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

Your Turn: Evaluate each expression and round to four decimal places. (a) Solution (a) (b)

Properties of Logarithms Examples Assume all variables are positive. Rewrite each expression using the properties of logarithms For x > 0, y > 0, a > 0, a  1, and any real number r, Product Rule Quotient Rule Power Rule

The Product Rule of Logarithms Product Rule of Logarithms If M, N, and a are positive real numbers, with a  1, then log a (MN) = log a M + log a N. (a) log 5 (4 · 7) log 5 (4 · 7) = log log 5 7 log 10 (100 · 1000) = log log = = 5 Example: Write the following logarithm as a sum of logarithms. (b) log 10 (100 · 1000)

Your Turn: Express as a sum of logarithms: Solution:

The Quotient Rule of Logarithms Quotient Rule of Logarithms If M, N, and a are positive real numbers, with a  1, then Example: Write the following logarithm as a difference of logarithms.

Your Turn: Express as a difference of logarithms. Solution:

Sum and Difference of Logarithms Example: Write as the sum or difference of logarithms. Quotient Rule Product Rule

The Power Rule of Logarithms Example: Use the Power Rule to express all powers as factors. log 4 (a 3 b 5 ) The Power Rule of Logarithms If M and a are positive real numbers, with a  1, and r is any real number, then log a M r = r log a M. = log 4 (a 3 ) + log 4 ( b 5 ) Product Rule = 3 log 4 a + 5 log 4 b Power Rule

Your Turn: Express as a product. Solution:

Your Turn: Express as a product. Solution:

NOT Laws of Logarithms Warning

Rewriting Logarithmic Expressions 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.

Examples: Expand: log 5mn = log 5 + log m + log n Expand: log 5 8x 3 = log ·log 5 x

Expand – Express as a Summ and Difference of 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

Condense - Express as a Single Logarithm Example: Write the following as the logarithm of a single expression. Power Rule Product Rule Quotient Rule

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

Examples: Condense: log ·log 5 t = log 5 7t 3 Condense: 3log 2 x – (log log 2 y)= log 2

Your Turn: Express in terms of sums and differences of logarithms. Solution:

Your Turn: Assume all variables are positive. Use the properties of logarithms to rewrite the expression Solution

Your Turn: Express as a single logarithm. Solution:

Your Turn: Use the properties of logarithms to write as a single logarithm with coefficient 1. Solution

Another Type of Problem If log a 3 = x and log a 4 = y, express each log expression in terms of x and y. 1.log a 12 Log a (3 4) = log a 3 + log a 4 = x+y 2.Log 3 4 Log 3 4 = log a 4/log a 3 = y/x

Assignment Pg. 211 – 213: #1 – 19 odd, 31, 33, 37 – 55 odd, 59 – 75 odd