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.

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: 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.

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 36 Simplify. Think: 6? = 36. Or convert to a base of 6 and solve for the exponent. 6 𝑦 = 6 2 log64 + log69 = 2

Express as a single logarithm. Simplify, if possible. log5625 + log525 Are the bases the same? To add the logarithms, multiply the numbers. log5 (625 • 25) Simplify. log5 15,625 Think: 5? = 15625 Convert to a base of 5 and solve for the exponent. 5 𝑦 = 5 6 log5625 + log525 = 6

( 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 27 + log 1 3 9 Are the bases the same? To add the logarithms, multiply the numbers. 1 3 log (27 • ) 9 Simplify. 1 3 log 3 Think: ? = 3 1 3 ( 1 3 ) 𝑦 = 3 1 Convert to a base of 1 3 and solve for the exponent. ( 1 3 ) 𝑦 =( 1 3 ) −1 log 1 3 27+ log 1 3 1 9 = –1

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.

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

log5100 – log54 = Express log5100 – log54 as a single logarithm. Simplify, if possible. log5100 – log54 Are the bases the same? To subtract the logarithms, divide the numbers. log5(100 ÷ 4) Simplify. log525 Think: 5? = 25. log5100 – log54 = 2

log 7 49− log 7 7= Express log749 – log77 as a single logarithm. Simplify, if possible. log749 – log77 Are the bases the same? To subtract the logarithms, divide the numbers log7(49 ÷ 7) Simplify. log77 Think: 7? = 7. log 7 49− log 7 7= 1

Because you can multiply logarithms, you can also take powers of logarithms.

6log232 = 30 Express as a product. Simplify, if possible. A. log2326 B. log8420 6log232 20log84 8 𝑦 =4 Because 25 = 32, log232 = 5. 2 3 𝑦 = 2 2 6(5) = 30 2 3𝑦 = 2 2 𝑦= 2 3 6log232 = 30 20( ) = 40 3 2 𝟒𝟎 𝟑 log 8 4 20 =

log104 =𝟒 log5252 =𝟒 Express as a product. Simplify, if possibly. Because 101 = 10, log 10 = 1. Because 52 = 25, log525 = 2. 4(1) = 4 2(2) = 4 log104 =𝟒 log5252 =𝟒

Solve log 2 1 2 2 𝑦 = 1 2 2 𝑦 = 2 −1 𝑦=−1 log 2 1 2 5 = −𝟓 Express as a product. Simplify, if possibly. log2 ( )5 1 2 Solve log 2 1 2 5log2 ( ) 1 2 2 𝑦 = 1 2 2 𝑦 = 2 −1 𝑦=−1 5(–1) = –5 log 2 1 2 5 = −𝟓

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) = log10100 + log101000 = 2 + 3 = 5

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

The 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.

Express 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 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 log4b Power Rule

Your Turn: Express as a product. Solution:

Your Turn: Express as a product. Solution:

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.

log 5mn = log 5 + log m + log n log58x3 = log58 + 3·log5x Examples: Expand: log 5mn = log 5 + log m + log n log58x3 = log58 + 3·log5x

Expand – Express as a Sum and Difference of Logarithms log27x3 - log2y = log27 + log2x3 – log2y = log27 + 3·log2x – log2y

Condensing Logarithms log 6 + 2 log2 – log 3 = log 6 + log 22 – log 3 = log (6·22) – log 3 = log = log 8

log57 + 3·log5t = log57t3 3log2x – (log24 + log2y)= log2 Examples: Condense: log57 + 3·log5t = log57t3 3log2x – (log24 + log2y)= log2

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

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 Formula If a  1, and b  1, and M are positive real numbers, then Example: Approximate log4 25. 10 is used for both bases.

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

Your Turn: Evaluate each expression and round to four decimal places. Solution (a) 1.7604 (b) -3.3219