<< 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

An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
Practice Key Terms 4

Get the best College algebra course in your pocket!





Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask