# Properties of Logarithms

These are my notes and worked examples of properties of logarithms.

Applications involving logarithms play an important role in modern day computation. One reason for this is that logarithms possess several important properties. For example, the loudness of a sound can be measured in decibels by the formula \(f(x) = 10\log (10^{16}x)\) where x is the intensity of the sound in watts per square centimeter.

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**Basic Properties of Logarithms**

Logarithms possess several important properties. One property of logarithms states that the sum of the logarithms of two numbers equals the logarithm of their product.

\[\log 5 + \log 2 = \log 10\]

\[\log 5 + \log 25 = \log 100\]

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**Property 1**

This property is a direct result of the inverse property \(\log_{a} a^{x} = x\)

\[\log_{a} 1 = \log_{a} a^{0} = 0\]

\[\log_{a} a = \log_{a} a^{1} = 1\]

So:

\[\log 1 = 0\]

\[\ln e = 1\]

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**Property 2**

If m and n are positive numbers, then we can write \(m = a^{x}\) and \(n = a^{d}\) for real numbers.

\[\log_{a} m + \log_{a} n = \log_{a} a^{x} + \log_{a} a^{d} = c + d\]

\[\log_{a}(mn) = \log_{a} (a^{x}a^{d}) = \log_{a} (a^{c + d}) = c + d\]

So:

\[\log_{a} m + \log_{a} n = \log_{a} (mn)\]

Example: Let \(m = 100 \text{ and } n = 1000\)

\[\log m \log n = \log 100 + \log 1000 = \log 10^{2} + \log 10^{3} = 2 + 3 = 5\]

\[\log (mn) = \log (100 * 1000) = \log 100000 = \log 10^{5} = 5\]

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**Property 3**

Let \(m = a^{c}\) and \(n = a^{d}\) for real numbers c and d

\[\log_{a} m - \log_{a} n = \log_{a} a^{c} - \log_{a} a^{d} = c - d\]

\[\log_{a} \frac{m}{n} = \log_{a} \frac{a^{c}}{a^{d}} = \log_{a} (a^{c - d}) = c - d\]

So:

\[\log_{a} m - \log_{a} n = \log_{a} \frac{m}{n}\]

Example: Let m = 100 and n = 1000

\[\log m - \log n = \log 100 - \log 1000 = \log 10^{2} - \log 10^{3} = 2 - 3 = -1\]

\[\log \frac{m}{n} = \log \frac{100}{1000} = \log \frac{1}{10} = \log 10^{-1} = -1\]

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**Property 4**

Let \(m = a^{c}\) and r be any real number

\[\log_{a} m^{r} = \log_{a} (a^{c})^{r} = \log_{a} (a^{cr}) = cr\]

\[r \log_{a} m = r \log_{a} a^{c} = rc\]

So:

\[\log_{a} m^{r} = r \log_{a} m\]

Example: Let m= 100 and r = 3

\[\log m^{r} = \log 100^{3} = \log 1,000,000 = \log 10^{6} = 6\]

\[r \log m = 3 \log 100 = 3 \log 10^{2} = 3 * 2 = 6\]

**Example**

Expand \(\log xy\)

\[\log xy = \log x + \log y\]

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**Example**

Expand \(\ln \frac{6}{z}\)

\[\ln \frac{6}{z} = \ln 6 - \ln z\]

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**Example**

Expand \(\log 2x^{4}\)

\[\log 2x^{4} = \log 2 + \log x^{4} = \log 2 + 4 \log x\]

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**Example**

Expand \(\ln \frac{7x^{3}}{k}\)

\[\ln \frac{7x^{3}}{k} = \ln 7x^{3} - \ln k = \ln 7 + \ln x^{3} - \ln k = \ln 7 + 3 \ln x - \ln k\]

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**Example**

Analyzing sound with decibels

Sound levels in decibels can be computed by \(D(x) = 10 \log (10^{16})x\)

Use properties of logarithms to simplify the formula

\[D(x) = 10 \log (10^{16})x\]

\[10(\log 10^{16}) + \log x\]

\[10(16 + \log x)\]

\[160 + 10 \log x\]

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**Example**

Write the expression as the logarithm of a single expression

\[\ln 2e + \ln \frac{1}{e}\]

\[\ln(2e*\frac{1}{e})\]

\[= \ln 2\]

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**Example**

Write the expression as the logarithm of a single expression

\[\log_{2} 27 + \log_{2} x^{3}\]

\[= \log_{2}(27x^{3})\]

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**Example**

Write the expression as the logarithm of a single expression

\[\log x^{3} - \log x^{2}\]

\[\log \frac{x^{3}}{x^{2}}\]

\[= \log x\]

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**Example**

Write the expression as the logarithm of a single expression

\[\log 5 + \log 15 - \log 3\]

\[\log (5 * 15) - \log 3\]

\[\log \frac{5 * 15}{3}\]

\[= \log 25\]

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**Example**

Write the expression as the logarithm of a single expression

\[2 \ln x - \frac{1}{2}\ln y - 3\ln z\]

\[\ln x^{2} - \ln y^{\frac{1}{2}} - \ln z^{3}\]

\[\ln \frac{x^{2}}{y^{\frac{1}{2}}} - \ln z^{3}\]

\[\ln \frac{x^{2}}{(y^{\frac{1}{2})}(z^{3})}\]

\[\ln \frac{x^{2}}{(z^{3})(\sqrt{y})}\]

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**Example**

Write the expression as the logarithm of a single expression

\[5 \log_{3} x + \log_{3} 2x - \log_{3} y\]

\[\log_{3} x^{5} + \log_{3} 2x - \log_{3} y\]

\[\log_{3} (x^{5} * 2x) - \log_{3} y\]

\[\log_{3} \frac{2x^{6}}{y}\]