4.4 Logarithmic properties (Page 5/10)

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Condensing complex logarithmic expressions

Condense $\log_{2} (x^{2}) + \frac{1}{2} \log_{2} (x - 1) - 3 \log_{2} ({(x + 3)}^{2}) .$

We apply the power rule first:

\log_{2} (x^{2}) + \frac{1}{2} \log_{2} (x - 1) - 3 \log_{2} ({(x + 3)}^{2}) = \log_{2} (x^{2}) + \log_{2} (\sqrt{x - 1}) - \log_{2} ({(x + 3)}^{6})

Next we apply the product rule to the sum:

\log_{2} (x^{2}) + \log_{2} (\sqrt{x - 1}) - \log_{2} ({(x + 3)}^{6}) = \log_{2} (x^{2} \sqrt{x - 1}) - \log_{2} ({(x + 3)}^{6})

Finally, we apply the quotient rule to the difference:

\log_{2} (x^{2} \sqrt{x - 1}) - \log_{2} ({(x + 3)}^{6}) = \log_{2} \frac{x^{2} \sqrt{x - 1}}{{(x + 3)}^{6}}

Rewriting as a single logarithm

Rewrite $2 \log x - 4 \log (x + 5) + \frac{1}{x} \log (3 x + 5)$ as a single logarithm.

We apply the power rule first:

\log (x + 5) + \frac{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 \frac{x^{2} {(3 x + 5)}^{x^{−1}}}{{(x + 5)}^{4}}

Try It

Rewrite $\log (5) + 0.5 \log (x) - \log (7 x - 1) + 3 \log (x - 1)$ as a single logarithm.

$\log (\frac{5 {(x - 1)}^{3} \sqrt{x}}{(7 x - 1)})$

Try It

Condense $4 (3 \log (x) + \log (x + 5) - \log (2 x + 3)) .$

$\log \frac{x^{12} {(x + 5)}^{4}}{{(2 x + 3)}^{4}};$ this answer could also be written $\log {(\frac{x^{3} (x + 5)}{(2 x + 3)})}^{4} .$

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) \approx P - 0.301

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

Try It

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 $n \neq 1$ and $b \neq 1,$ we show

\log_{b} M = \frac{\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

\begin{array}{l} \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 & = \frac{\log_{n} M}{\log_{n} b} & Isolate y . \\ \log_{b} M & = \frac{\log_{n} M}{\log_{n} b} & Substitute for y . \end{array}

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.

\begin{array}{l} \log_{5} 36 & = \frac{\log (36)}{\log (5)} & Apply the change of base formula using base 10 . \\ \approx 2.2266 & Use a calculator to evaluate to 4 decimal places . \end{array}

A General Note

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 $n \neq 1$ and $b \neq 1,$

\log_{b} M = \frac{\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 = \frac{\ln M}{\ln b}

and

\log_{b} M = \frac{\log M}{\log b}

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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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