# Properties of Logarithms

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16-Feb-2016Category

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Properties of Logarithms

Properties of LogarithmsSection 3.3ObjectivesRewrite logarithms with different bases.Use properties of logarithms to evaluate or rewrite logarithmic expressions.Use properties of logarithms to expand or condense logarithmic expressions.

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.Logarithmic Notation For logarithmic functions we use the notation: loga(x) or logax

This is read log, base a, of x. Thus, y = logax means x = ay

And so a logarithm is simply an exponent of some base.

Remember that to multiply powers with the same base, you add exponents.

Express log64 + log69 as a single logarithm.Simplify.Adding Logarithms2To add the logarithms, multiply the numbers.log6 (4 9)log6 36Simplify.Think: 6? = 36.log64 + log69 =Or convert to a base of 6 and solve for the exponent.Are the bases the same?6Express as a single logarithm.Simplify, if possible.6To add the logarithms, multiply the numbers.log5625 + log525log5 (625 25)log5 15,625Simplify.Think: 5? = 15625Convert to a base of 5 and solve for the exponent.log5625 + log525 =Are the bases the same?7Express as a single logarithm.Simplify, if possible.1To add the logarithms, multiply the numbers.Simplify.Think: ? = 31313 log (27 )1913log 3log 27 + log131319Are the bases the same?8Remember 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. 9The property above can also be used in reverse.

Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified. Caution10Express log5100 log54 as a single logarithm. Simplify, if possible.To subtract the logarithms, divide the numbers.log5100 log54log5(100 4)2log525Simplify.Think: 5? = 25.Are the bases the same?log5100 log54 =11Express log749 log77 as a single logarithm.Simplify, if possible.To subtract the logarithms, divide the numberslog749 log77log7(49 7)1log77Simplify.Think: 7? = 7.Are the bases the same?12Because you can multiply logarithms, you can also take powers of logarithms.

13Express as a product.Simplify, if possible.A. log2326B. log84206log2326(5) = 3020log8420( ) = 40 3 2 3Because 25 = 32, log232 = 5. 6log232 =3014Express as a product.Simplify, if possibly.log104log52524log104(1) = 42log5252(2) = 4Because 52 = 25, log525 = 2.Because 101 = 10, log 10 = 1. log104log525215Express as a product.Simplify, if possibly. log2 ( )55(1) = 55log2 ( ) 1 2 1 216The Product Rule of LogarithmsProduct Rule of LogarithmsIf M, N, and a are positive real numbers, with a 1, then loga(MN) = logaM + logaN.(a) log5(4 7)log5(4 7) = log54 + log57log10(100 1000) = log10100 + log101000= 2 + 3 = 5 Example: Write the following logarithm as a sum of logarithms.(b) log10(100 1000)17Your Turn:Express as a sum of logarithms:

Solution:

The Quotient Rule of LogarithmsQuotient Rule of LogarithmsIf M, N, and a are positive real numbers, with a 1, thenExample: Write the following logarithm as a difference of logarithms.

19Your Turn:Express as a difference of logarithms.

Solution:

Sum and Difference of LogarithmsExample: Write as the sum or difference of logarithms.

Quotient Rule

Product Rule21The Power Rule of LogarithmsExample: Use the Power Rule to express all powers as factors. log4(a3b5)The Power Rule of LogarithmsIf M and a are positive real numbers, with a 1, and r is any real number, then loga M r = r loga M.= log4(a3) + log4(b5)Product Rule= 3 log4a + 5 log4bPower Rule22Your Turn:Express as a product.

Solution:

Your Turn:Express as a product.

Solution:

Rewriting Logarithmic ExpressionsThe 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:log58x3 =log58 + 3log5x

Expand Express as a Sum and Difference of Logarithmslog2 =

log27x3 - log2y = log27 + log2x3 log2y =log27 + 3log2x log2y

Condensing Logarithmslog 6 + 2 log2 log 3 =log 6 + log 22 log 3 =log (622) log 3 =

log =

log 8

Examples:Condense:log57 + 3log5t =log57t3

Condense:3log2x (log24 + log2y)=

log2

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

Solution:

Change-of-Base FormulaExample:Approximate log4 25.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, then

10 is used for both bases.Change-of-Base FormulaExample:Approximate the following logarithms.

Your Turn:Evaluate each expression and round to four decimal places.

Solution

(a) 1.7604

(b) -3.3219

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