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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

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

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 ) )

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 .

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.

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

The pH increases by about 0.301.

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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
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Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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