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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

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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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