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Given a logarithm with the form log b M , use the change-of-base formula to rewrite it as a quotient of logs with any positive base n , where n 1.

  1. Determine the new base n , remembering that the common log, log ( x ) , has base 10, and the natural log, ln ( x ) , has base e .
  2. Rewrite the log as a quotient using the change-of-base formula
    • The numerator of the quotient will be a logarithm with base n and argument M .
    • The denominator of the quotient will be a logarithm with base n and argument b .

Changing logarithmic expressions to expressions involving only natural logs

Change log 5 3 to a quotient of natural logarithms.

Because we will be expressing log 5 3 as a quotient of natural logarithms, the new base, n = e .

We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.

log b M = ln M ln b log 5 3 = ln 3 ln 5
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Change log 0.5 8 to a quotient of natural logarithms.

ln 8 ln 0.5

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Can we change common logarithms to natural logarithms?

Yes. Remember that log 9 means log 10 9 . So, log 9 = ln 9 ln 10 .

Using the change-of-base formula with a calculator

Evaluate log 2 ( 10 ) using the change-of-base formula with a calculator.

According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base e .

log 2 10 = ln 10 ln 2 Apply the change of base formula using base  e . 3.3219 Use a calculator to evaluate to 4 decimal places .
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Evaluate log 5 ( 100 ) using the change-of-base formula.

ln 100 ln 5 4.6051 1.6094 = 2.861

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Access these online resources for additional instruction and practice with laws of logarithms.

Key equations

The Product Rule for Logarithms log b ( M N ) = log b ( M ) + log b ( N )
The Quotient Rule for Logarithms log b ( M N ) = log b M log b N
The Power Rule for Logarithms log b ( M n ) = n log b M
The Change-of-Base Formula log b M = log n M log n b           n > 0 , n 1 , b 1

Key concepts

  • We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See [link] .
  • We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See [link] .
  • We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See [link] , [link] , and [link] .
  • We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See [link] , [link] , and [link] .
  • The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See [link] , [link] , [link] , and [link] .
  • We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See [link] .
  • The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and e as the quotient of natural or common logs. That way a calculator can be used to evaluate. See [link] .

Section exercises

Verbal

How does the power rule for logarithms help when solving logarithms with the form log b ( x n ) ?

Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, log b ( x 1 n ) = 1 n log b ( x ) .

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What does the change-of-base formula do? Why is it useful when using a calculator?

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Algebraic

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

log b ( 7 x 2 y )

log b ( 2 ) + log b ( 7 ) + log b ( x ) + log b ( y )

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log b ( 13 17 )

log b ( 13 ) log b ( 17 )

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ln ( 1 4 k )

k ln ( 4 )

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For the following exercises, condense to a single logarithm if possible.

ln ( 7 ) + ln ( x ) + ln ( y )

ln ( 7 x y )

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log 3 ( 2 ) + log 3 ( a ) + log 3 ( 11 ) + log 3 ( b )

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log b ( 28 ) log b ( 7 )

log b ( 4 )

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ln ( a ) ln ( d ) ln ( c )

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log b ( 1 7 )

log b ( 7 )

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For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

log ( x 15 y 13 z 19 )

15 log ( x ) + 13 log ( y ) 19 log ( z )

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ln ( a −2 b −4 c 5 )

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log ( x 3 y 4 )

3 2 log ( x ) 2 log ( y )

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log ( x 2 y 3 x 2 y 5 3 )

8 3 log ( x ) + 14 3 log ( y )

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For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

log ( 2 x 4 ) + log ( 3 x 5 )

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ln ( 6 x 9 ) ln ( 3 x 2 )

ln ( 2 x 7 )

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

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

log ( x z 3 y )

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4 log 7 ( c ) + log 7 ( a ) 3 + log 7 ( b ) 3

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For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

log 7 ( 15 ) to base e

log 7 ( 15 ) = ln ( 15 ) ln ( 7 )

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log 14 ( 55.875 ) to base 10

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For the following exercises, suppose log 5 ( 6 ) = a and log 5 ( 11 ) = b . Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and b . Show the steps for solving.

log 11 ( 5 )

log 11 ( 5 ) = log 5 ( 5 ) log 5 ( 11 ) = 1 b

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log 11 ( 6 11 )

log 11 ( 6 11 ) = log 5 ( 6 11 ) log 5 ( 11 ) = log 5 ( 6 ) log 5 ( 11 ) log 5 ( 11 ) = a b b = a b 1

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Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.

log 3 ( 1 9 ) 3 log 3 ( 3 )

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6 log 8 ( 2 ) + log 8 ( 64 ) 3 log 8 ( 4 )

3

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2 log 9 ( 3 ) 4 log 9 ( 3 ) + log 9 ( 1 729 )

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For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

log 1 2 ( 4.7 )

2.23266

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Extensions

Use the product rule for logarithms to find all x values such that log 12 ( 2 x + 6 ) + log 12 ( x + 2 ) = 2. Show the steps for solving.

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Use the quotient rule for logarithms to find all x values such that log 6 ( x + 2 ) log 6 ( x 3 ) = 1. Show the steps for solving.

x = 4 ; By the quotient rule: log 6 ( x + 2 ) log 6 ( x 3 ) = log 6 ( x + 2 x 3 ) = 1.

Rewriting as an exponential equation and solving for x :

6 1 = x + 2 x 3 0 = x + 2 x 3 6 0 = x + 2 x 3 6 ( x 3 ) ( x 3 ) 0 = x + 2 6 x + 18 x 3 0 = x 4 x 3 x = 4

Checking, we find that log 6 ( 4 + 2 ) log 6 ( 4 3 ) = log 6 ( 6 ) log 6 ( 1 ) is defined, so x = 4.

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Can the power property of logarithms be derived from the power property of exponents using the equation b x = m ? If not, explain why. If so, show the derivation.

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Prove that log b ( n ) = 1 log n ( b ) for any positive integers b > 1 and n > 1.

Let b and n be positive integers greater than 1. Then, by the change-of-base formula, log b ( n ) = log n ( n ) log n ( b ) = 1 log n ( b ) .

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Does log 81 ( 2401 ) = log 3 ( 7 ) ? Verify the claim algebraically.

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Questions & Answers

So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
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Joe
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Rima
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Joe
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Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
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mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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