2 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.
3 Logarithmic FAQsLogarithms 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.
4 Logarithmic Notation For logarithmic functions we use the notation: loga(x) or logaxThis is read “log, base a, of x.” Thus,y = logax means x = ayAnd so a logarithm is simply an exponent of some base.
5 Remember that to multiply powers with the same base, you add exponents.
6 Are the bases the same? 6 𝑦 = 6 2 log64 + log69 = Adding Logarithms Express log64 + log69 as a single logarithm.Simplify.Are the bases the same?To add the logarithms, multiply the numbers.log6 (4 9)log6 36Simplify.Think: 6? = 36.Or convert to a base of 6 and solve for the exponent.6 𝑦 = 6 2log64 + log69 =2
7 Express as a single logarithm. Simplify, if possible.log log525Are the bases the same?To add the logarithms, multiply the numbers.log5 (625 • 25)Simplify.log5 15,625Think: 5? = 15625Convert to a base of 5 and solve for the exponent.5 𝑦 = 5 6log log525 =6
8 ( 1 3 ) 𝑦 =( 1 3 ) −1 ( 1 3 ) 𝑦 = 3 1 log 1 3 27+ log 1 3 1 9 = –1 Express as a single logarithm.Simplify, if possible.log log139Are the bases the same?To add the logarithms, multiply the numbers.13log (27 • )9Simplify.13log 3Think: ? = 313( 1 3 ) 𝑦 = 3 1Convert to a base of and solve for the exponent.( 1 3 ) 𝑦 =( 1 3 ) −1log log =–1
9 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.
10 The property above can also be used in reverse. Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.Caution
11 log5100 – log54 = Express log5100 – log54 as a single logarithm. Simplify, if possible.log5100 – log54Are the bases the same?To subtract the logarithms, divide the numbers.log5(100 ÷ 4)Simplify.log525Think: 5? = 25.log5100 – log54 =2
12 log 7 49− log 7 7= Express log749 – log77 as a single logarithm. Simplify, if possible.log749 – log77Are the bases the same?To subtract the logarithms, divide the numberslog7(49 ÷ 7)Simplify.log77Think: 7? = 7.log 7 49− log 7 7=1
13 Because you can multiply logarithms, you can also take powers of logarithms.
14 6log232 = 30 Express as a product. Simplify, if possible. A. log2326 B. log84206log23220log848 𝑦 =4Because 25 = 32, log232 = 5.2 3 𝑦 = 2 26(5) = 302 3𝑦 = 2 2𝑦= 2 36log232 =3020( ) =4032𝟒𝟎 𝟑log =
15 log104 =𝟒 log5252 =𝟒 Express as a product. Simplify, if possibly. Because = 10, log 10 = 1.Because = 25, log525 = 2.4(1) = 42(2) = 4log104=𝟒log5252=𝟒
17 The Product Rule of Logarithms If M, N, and a are positive real numbers, with a 1, then loga(MN) = logaM + logaN.Example: Write the following logarithm as a sum of logarithms.(a) log5(4 · 7)log5(4 · 7) = log54 + log57(b) log10(100 · 1000)log10(100 · 1000) = log log101000= = 5
18 Express as a sum of logarithms: Your Turn:Express as a sum of logarithms:Solution:
19 The Quotient Rule of Logarithms If M, N, and a are positive real numbers, with a 1, thenExample: Write the following logarithm as a difference of logarithms.
20 Express as a difference of logarithms. Your Turn:Express as a difference of logarithms.Solution:
21 Sum and Difference of Logarithms Example: Write as the sum or difference of logarithms.Quotient RuleProduct Rule
22 The Power Rule of Logarithms If M and a are positive real numbers, with a 1, and r is any real number, then loga M r = r loga M.Example: Use the Power Rule to express all powers as factors.log4(a3b5)= log4(a3) + log4(b5)Product Rule= 3 log4a + 5 log4bPower Rule
25 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.
30 Your Turn: Express in terms of sums and differences of logarithms. Solution:
31 Change-of-Base Formula 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 FormulaIf a 1, and b 1, and M are positive real numbers, thenExample:Approximate log4 25.10 is used for both bases.
32 Change-of-Base Formula Example:Approximate the following logarithms.
33 Your Turn: Evaluate each expression and round to four decimal places. Solution(a)(b)