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What Are the Logarithm Rules?

As you can see below, all the logarithmic rules for the common logarithm $\log x$ and the natural logarithm $\ln x$ are the same:

Common logarithm:: $1 0^{\log a} = a$
Natural logarithm:: $e^{\ln a} = a$

This is true for any logarithm regardless of its base.

Rule

The Logarithmic Rules for the Common Logarithm

The first logarithmic rule:

\log (a^{x}) = x \log a

The second logarithmic rule:

\log (a \cdot b) = \log a + \log b

The third logarithmic rule:

\log (\frac{a}{b}) = \log a - \log b

Rule

The Logarithmic Rules for the Natural Logarithm

The first logarithmic rule

\ln (a^{x}) = x \ln a

The second logarithmic rule

\ln (a \cdot b) = \ln a + \ln b

The third logarithmic rule

\ln (\frac{a}{b}) = \ln a - \ln b

Example 1

Simplify $\log a^{2} + \log b^{2} - 2 \log a$

$\begin{array}{l} \log a^{2} + \log b^{2} - 2 \log a & = 2 \log a + 2 \log b \\ - 2 \log a \\ = 2 \log b \end{array}$

$\begin{array}{l} \log a^{2} + \log b^{2} - 2 \log a & = 2 \log a + 2 \log b - 2 \log a \\ = 2 \log b \end{array}$

Example 2

Simplify $\log a b + \log b^{2} - \log a^{2} b$

$\begin{array}{l} \log a b + \log b^{2} - \log a^{2} b & = \log a + \log b + 2 \log b \\ - (\log a^{2} + \log b) \\ = \log a + 3 \log b - \log a^{2} \\ - \log b \\ = \log a + 2 \log b \\ - 2 \log a \\ = - \log a + 2 \log b \end{array}$

$\begin{array}{l} \log a b + \log b^{2} - \log a^{2} b & = \log a + \log b + 2 \log b - (\log a^{2} + \log b) \\ = \log a + 3 \log b - \log a^{2} - \log b \\ = \log a + 2 \log b - 2 \log a \\ = - \log a + 2 \log b \end{array}$

Example 3

Simplify $\log \frac{a}{b} - \log \frac{2 a}{b^{3}}$

\begin{array}{l} \log \frac{a}{b} - \log \frac{2 a}{b^{3}} & = \log a - \log b - (\log 2 a - \log b^{3}) \\ = \log a - \log b - (\log 2 + \log a \\ - 3 \log b) \\ = \log a - \log b - \log 2 - \log a \\ + 3 \log b \\ = 2 \log b - \log 2 \end{array}

\begin{array}{l} \log \frac{a}{b} - \log \frac{2 a}{b^{3}} & = \log a - \log b - (\log 2 a - \log b^{3}) \\ = \log a - \log b - (\log 2 + \log a - 3 \log b) \\ = \log a - \log b - \log 2 - \log a + 3 \log b \\ = 2 \log b - \log 2 \end{array}

Example 4

Simplify $\log 2 x + \log 2 - \log \frac{2}{x^{2}} + \log 10$

\begin{array}{l} \log 2 x + \log 2 - \log \frac{2}{x^{2}} + \log 10 & = \log 2 + \log x \\ + \log 2 - (\log 2 \\ - \log x^{2}) + 1 \\ = 2 \log 2 + \log x \\ - \log 2 + \log x^{2} \\ + 1 \\ = \log 2 + \log x \\ + 2 \log x + 1 \\ = \log 2 + 3 \log x \\ + 1 \end{array}

\begin{array}{l} \log 2 x + \log 2 - \log \frac{2}{x^{2}} + \log 10 & = \log 2 + \log x + \log 2 - (\log 2 - \log x^{2}) + 1 \\ = 2 \log 2 + \log x - \log 2 + \log x^{2} + 1 \\ = \log 2 + \log x + 2 \log x + 1 \\ = \log 2 + 3 \log x + 1 \end{array}

Example 5

Use the logarithmic rules to simplify $\ln 2 x - \ln (\frac{x}{2}) - 4 \ln x$

\begin{array}{l} \ln 2 x - \ln (\frac{x}{2}) - 4 \ln x \\ = \ln 2 + \ln x - (\ln x - \ln 2) - 4 \ln x \\ = \ln 2 + \ln x - \ln x + \ln 2 - 4 \ln x \\ = 2 \ln 2 - 4 \ln x \end{array}

\begin{array}{l} \ln 2 x - \ln (\frac{x}{2}) - 4 \ln x & = \ln 2 + \ln x - (\ln x - \ln 2) - 4 \ln x \\ = \ln 2 + \ln x - \ln x + \ln 2 - 4 \ln x \\ = 2 \ln 2 - 4 \ln x \end{array}

Example 6

Use the logarithmic rules to simplify $\ln 2 x^{3} - \ln (\frac{3 x}{2}) + \ln {(3 x)}^{2}$

\begin{array}{l} \ln 2 x^{3} - \ln (\frac{3 x}{2}) + \ln {(3 x)}^{2} \\ = \ln 2 + \ln x^{3} - (\ln 3 x - \ln 2) + \ln 3^{2} x^{2} \\ = \ln 2 + 3 \ln x - (\ln 3 + \ln x - \ln 2) + \ln 3^{2} \\ + \ln x^{2} \\ = \ln 2 + 3 \ln x - \ln 3 - \ln x + \ln 2 + 2 \ln 3 \\ + 2 \ln x \\ = 2 \ln 2 + 4 \ln x + \ln 3 \end{array}

\begin{array}{l} \ln 2 x^{3} - \ln (\frac{3 x}{2}) + \ln {(3 x)}^{2} & = \ln 2 + \ln x^{3} - (\ln 3 x - \ln 2) + \ln 3^{2} x^{2} \\ = \ln 2 + 3 \ln x - (\ln 3 + \ln x - \ln 2) + \ln 3^{2} \\ + \ln x^{2} \\ = \ln 2 + 3 \ln x - \ln 3 - \ln x + \ln 2 + 2 \ln 3 \\ + 2 \ln x \\ = 2 \ln 2 + 4 \ln x + \ln 3 \end{array}

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