6.5 Logarithmic properties (Page 2/10)

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\log_{b} (M N) = \log_{b} (M) + \log_{b} (N) .

Let $m = \log_{b} M$ and $n = \log_{b} N .$ In exponential form, these equations are $b^{m} = M$ and $b^{n} = N .$ It follows that

\begin{array}{l} \log_{b} (M N) & = \log_{b} (b^{m} b^{n}) & Substitute for M and N . \\ = \log_{b} (b^{m + n}) & Apply the product rule for exponents . \\ = m + n & Apply the inverse property of logs . \\ = \log_{b} (M) + \log_{b} (N) & Substitute for m and n . \end{array}

Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider $\log_{b} (w x y z) .$ Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:

\log_{b} (w x y z) = \log_{b} w + \log_{b} x + \log_{b} y + \log_{b} z

A General Note

The product rule for logarithms

The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.

\log_{b} (M N) = \log_{b} (M) + \log_{b} (N) for b > 0

How To

Given the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.

Factor the argument completely, expressing each whole number factor as a product of primes.
Write the equivalent expression by summing the logarithms of each factor.

Using the product rule for logarithms

Expand $\log_{3} (30 x (3 x + 4)) .$

We begin by factoring the argument completely, expressing $30$ as a product of primes.

\log_{3} (30 x (3 x + 4)) = \log_{3} (2 \cdot 3 \cdot 5 \cdot x \cdot (3 x + 4))

Next we write the equivalent equation by summing the logarithms of each factor.

\log_{3} (30 x (3 x + 4)) = \log_{3} (2) + \log_{3} (3) + \log_{3} (5) + \log_{3} (x) + \log_{3} (3 x + 4)

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Expand $\log_{b} (8 k) .$

$\log_{b} 2 + \log_{b} 2 + \log_{b} 2 + \log_{b} k = 3 \log_{b} 2 + \log_{b} k$

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Using the quotient rule for logarithms

For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting: $x^{\frac{a}{b}} = x^{a - b} .$ The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.

Given any real number $x$ and positive real numbers $M,$ $N,$ and $b,$ where $b \neq 1,$ we will show

\log_{b} (\frac{M}{N}) = \log_{b} (M) - \log_{b} (N) .

Let $m = \log_{b} M$ and $n = \log_{b} N .$ In exponential form, these equations are $b^{m} = M$ and $b^{n} = N .$ It follows that

\begin{array}{l} \log_{b} (\frac{M}{N}) & = \log_{b} (\frac{b^{m}}{b^{n}}) & Substitute for M and N . \\ = \log_{b} (b^{m - n}) & Apply the quotient rule for exponents . \\ = m - n & Apply the inverse property of logs . \\ = \log_{b} (M) - \log_{b} (N) & Substitute for m and n . \end{array}

For example, to expand $\log (\frac{2 x^{2} + 6 x}{3 x + 9}),$ we must first express the quotient in lowest terms. Factoring and canceling we get,

\begin{array}{l} \log (\frac{2 x^{2} + 6 x}{3 x + 9}) = \log (\frac{2 x (x + 3)}{3 (x + 3)}) & Factor the numerator and denominator . \\ = \log (\frac{2 x}{3}) & Cancel the common factors . \end{array}

Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.

\begin{array}{l} \log (\frac{2 x}{3}) = \log (2 x) - \log (3) \\ = \log (2) + \log (x) - \log (3) \end{array}

A General Note

The quotient rule for logarithms

The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.

\log_{b} (\frac{M}{N}) = \log_{b} M - \log_{b} N

How To

Given the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.

Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.
Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.
Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

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Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

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BenJay

Method

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Rood

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Sir

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Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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