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Condensing complex logarithmic expressions

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

We apply the power rule first:

log 2 ( x 2 ) + 1 2 log 2 ( x 1 ) 3 log 2 ( ( x + 3 ) 2 ) = log 2 ( x 2 ) + log 2 ( x 1 ) log 2 ( ( x + 3 ) 6 )

Next we apply the product rule to the sum:

log 2 ( x 2 ) + log 2 ( x 1 ) log 2 ( ( x + 3 ) 6 ) = log 2 ( x 2 x 1 ) log 2 ( ( x + 3 ) 6 )

Finally, we apply the quotient rule to the difference:

log 2 ( x 2 x 1 ) log 2 ( ( x + 3 ) 6 ) = log 2 x 2 x 1 ( x + 3 ) 6
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Rewriting as a single logarithm

Rewrite 2 log x 4 log ( x + 5 ) + 1 x log ( 3 x + 5 ) as a single logarithm.

We apply the power rule first:

log ( x + 5 ) + 1 x log ( 3 x + 5 ) = log ( x 2 ) log ( x + 5 ) 4 + log ( ( 3 x + 5 ) x 1 )

Next we rearrange and apply the product rule to the sum:

log ( x 2 ) log ( x + 5 ) 4 + log ( ( 3 x + 5 ) x 1 )
= log ( x 2 ) + log ( ( 3 x + 5 ) x 1 ) log ( x + 5 ) 4
= log ( x 2 ( 3 x + 5 ) x 1 ) log ( x + 5 ) 4

Finally, we apply the quotient rule to the difference:

= log ( x 2 ( 3 x + 5 ) x −1 ) log ( x + 5 ) 4 = log x 2 ( 3 x + 5 ) x −1 ( x + 5 ) 4
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Rewrite log ( 5 ) + 0.5 log ( x ) log ( 7 x 1 ) + 3 log ( x 1 ) as a single logarithm.

log ( 5 ( x 1 ) 3 x ( 7 x 1 ) )

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

log x 12 ( x + 5 ) 4 ( 2 x + 3 ) 4 ; this answer could also be written log ( x 3 ( x + 5 ) ( 2 x + 3 ) ) 4 .

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Applying of the laws of logs

Recall that, in chemistry, pH = log [ H + ] . If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH?

Suppose C is the original concentration of hydrogen ions, and P is the original pH of the liquid. Then P = log ( C ) . If the concentration is doubled, the new concentration is 2 C . Then the pH of the new liquid is

pH = log ( 2 C )

Using the product rule of logs

pH = log ( 2 C ) = ( log ( 2 ) + log ( C ) ) = log ( 2 ) log ( C )

Since P = log ( C ) , the new pH is

pH = P log ( 2 ) P 0.301

When the concentration of hydrogen ions is doubled, the pH decreases by about 0.301.

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How does the pH change when the concentration of positive hydrogen ions is decreased by half?

The pH increases by about 0.301.

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Using the change-of-base formula for logarithms

Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or e , we use the change-of-base formula    to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.

To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms    .

Given any positive real numbers M , b , and n , where n 1   and b 1 , we show

log b M = log n M log n b

Let y = log b M . By taking the log base n of both sides of the equation, we arrive at an exponential form, namely b y = M . It follows that

log n ( b y ) = log n M Apply the one-to-one property . y log n b = log n M   Apply the power rule for logarithms . y = log n M log n b Isolate  y . log b M = log n M log n b Substitute for  y .

For example, to evaluate log 5 36 using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

log 5 36 = log ( 36 ) log ( 5 ) Apply the change of base formula using base 10 . 2.2266   Use a calculator to evaluate to 4 decimal places .

The change-of-base formula

The change-of-base formula    can be used to evaluate a logarithm with any base.

For any positive real numbers M , b , and n , where n 1   and b 1 ,

log b M = log n M log n b .

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.

log b M = ln M ln b

and

log b M = log M log b

Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
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Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
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Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb
Practice Key Terms 4

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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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