<< Chapter < Page Chapter >> Page >

Using the quotient rule for logarithms

Expand log 2 ( 15 x ( x 1 ) ( 3 x + 4 ) ( 2 x ) ) .

First we note that the quotient is factored and in lowest terms, so we apply the quotient rule.

log 2 ( 15 x ( x 1 ) ( 3 x + 4 ) ( 2 x ) ) = log 2 ( 15 x ( x 1 ) ) log 2 ( ( 3 x + 4 ) ( 2 x ) )

Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that the prime factors of the factor 15 are 3 and 5.

log 2 ( 15 x ( x 1 ) ) log 2 ( ( 3 x + 4 ) ( 2 x ) ) = [ log 2 ( 3 ) + log 2 ( 5 ) + log 2 ( x ) + log 2 ( x 1 ) ] [ log 2 ( 3 x + 4 ) + log 2 ( 2 x ) ]                                                                   = log 2 ( 3 ) + log 2 ( 5 ) + log 2 ( x ) + log 2 ( x 1 ) log 2 ( 3 x + 4 ) log 2 ( 2 x )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Expand log 3 ( 7 x 2 + 21 x 7 x ( x 1 ) ( x 2 ) ) .

log 3 ( x + 3 ) log 3 ( x 1 ) log 3 ( x 2 )

Got questions? Get instant answers now!

Using the power rule for logarithms

We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as x 2 ? One method is as follows:

log b ( x 2 ) = log b ( x x ) = log b x + log b x = 2 log b x

Notice that we used the product rule for logarithms    to find a solution for the example above. By doing so, we have derived the power rule for logarithms , which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,

100 = 10 2 3 = 3 1 2 1 e = e 1

The power rule for logarithms

The power rule for logarithms    can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

log b ( M n ) = n log b M

Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.

  1. Express the argument as a power, if needed.
  2. Write the equivalent expression by multiplying the exponent times the logarithm of the base.

Expanding a logarithm with powers

Expand log 2 x 5 .

The argument is already written as a power, so we identify the exponent, 5, and the base, x , and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

log 2 ( x 5 ) = 5 log 2 x
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Expand ln x 2 .

2 ln x

Got questions? Get instant answers now!

Rewriting an expression as a power before using the power rule

Expand log 3 ( 25 ) using the power rule for logs.

Expressing the argument as a power, we get log 3 ( 25 ) = log 3 ( 5 2 ) .

Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

log 3 ( 5 2 ) = 2 log 3 ( 5 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Expand ln ( 1 x 2 ) .

2 ln ( x )

Got questions? Get instant answers now!

Using the power rule in reverse

Rewrite 4 ln ( x ) using the power rule for logs to a single logarithm with a leading coefficient of 1.

Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression 4 ln ( x ) , we identify the factor, 4, as the exponent and the argument, x , as the base, and rewrite the product as a logarithm of a power: 4 ln ( x ) = ln ( x 4 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Rewrite 2 log 3 4 using the power rule for logs to a single logarithm with a leading coefficient of 1.

log 3 16

Got questions? Get instant answers now!

Expanding logarithmic expressions

Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example:

Questions & Answers

how fast can i understand functions without much difficulty
Joe Reply
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask