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log b ( 6 x y ) = log b ( 6 x ) log b y = log b 6 + log b x log b y

We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:

log b ( A C ) = log b ( A C 1 ) = log b ( A ) + log b ( C 1 ) = log b A + ( 1 ) log b C = log b A log b C

We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.

With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.

Expanding logarithms using product, quotient, and power rules

Rewrite ln ( x 4 y 7 ) as a sum or difference of logs.

First, because we have a quotient of two expressions, we can use the quotient rule:

ln ( x 4 y 7 ) = ln ( x 4 y ) ln ( 7 )

Then seeing the product in the first term, we use the product rule:

ln ( x 4 y ) ln ( 7 ) = ln ( x 4 ) + ln ( y ) ln ( 7 )

Finally, we use the power rule on the first term:

ln ( x 4 ) + ln ( y ) ln ( 7 ) = 4 ln ( x ) + ln ( y ) ln ( 7 )
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Expand log ( x 2 y 3 z 4 ) .

2 log x + 3 log y 4 log z

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Using the power rule for logarithms to simplify the logarithm of a radical expression

Expand log ( x ) .

log ( x ) = log x ( 1 2 ) = 1 2 log x
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Expand ln ( x 2 3 ) .

2 3 ln x

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Can we expand ln ( x 2 + y 2 ) ?

No. There is no way to expand the logarithm of a sum or difference inside the argument of the logarithm.

Expanding complex logarithmic expressions

Expand log 6 ( 64 x 3 ( 4 x + 1 ) ( 2 x 1 ) ) .

We can expand by applying the Product and Quotient Rules.

log 6 ( 64 x 3 ( 4 x + 1 ) ( 2 x 1 ) ) = log 6 64 + log 6 x 3 + log 6 ( 4 x + 1 ) log 6 ( 2 x 1 ) Apply the Quotient Rule . = log 6 2 6 + log 6 x 3 + log 6 ( 4 x + 1 ) log 6 ( 2 x 1 ) Simplify by writing  64 as 2 6 . = 6 log 6 2 + 3 log 6 x + log 6 ( 4 x + 1 ) log 6 ( 2 x 1 ) Apply the Power Rule .
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Expand ln ( ( x 1 ) ( 2 x + 1 ) 2 ( x 2 9 ) ) .

1 2 ln ( x 1 ) + ln ( 2 x + 1 ) ln ( x + 3 ) ln ( x 3 )

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Condensing logarithmic expressions

We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.

Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm.

  1. Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.
  2. Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.
  3. Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.

Using the product and quotient rules to combine logarithms

Write log 3 ( 5 ) + log 3 ( 8 ) log 3 ( 2 ) as a single logarithm.

Using the product and quotient rules

log 3 ( 5 ) + log 3 ( 8 ) = log 3 ( 5 8 ) = log 3 ( 40 )

This reduces our original expression to

log 3 ( 40 ) log 3 ( 2 )

Then, using the quotient rule

log 3 ( 40 ) log 3 ( 2 ) = log 3 ( 40 2 ) = log 3 ( 20 )
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Condense log 3 log 4 + log 5 log 6.

log ( 3 5 4 6 ) ; can also be written log ( 5 8 ) by reducing the fraction to lowest terms.

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Questions & Answers

how to understand calculus?
Jenica Reply
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
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
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra 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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