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Properties of Logarithms

Use properties to expand or condense logarithms.

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Properties of Logarithms

What is the value of the expression \begin{align*}log_6 (8) + log_6 (27)\end{align*}?

Alone, neither of these expressions has an integer value, therefore combining them might seem like a bit of a challenge. The value of log6 8 is between 1 and 2; the value of log6 27 is also between 1 and 2.

Is there an easier way?

Properties of Logarithms

Previously we defined the logarithmic function as the inverse of an exponential function, and we evaluated log expressions in order to identify values of these functions. In this lesson we will work with more complicated log expressions. We will use the properties of logarithms to write a log expression as the sum or difference of several expressions, or to write several expressions as a single log expression.

Properties of Logarithms

Because a logarithm is an exponent, the properties of logs reflect the properties of exponents.

The basic properties are:

  • \begin{align*}log_b (xy) = log_b x + log_b y\end{align*}
  • \begin{align*}log_b \left (\frac{x} {y}\right ) = log_b x - log_b y\end{align*}
  • \begin{align*}log_b x^{n} = n log_b x\end{align*}

Expanding Expressions

Using the properties of logs, we can write a log expression as the sum or difference of simpler expressions. Consider the following examples:

  1. \begin{align*}log_2 8x\end{align*} = \begin{align*}log_2 8 + log_2 x\end{align*} = \begin{align*}3 + log_2 x\end{align*}
  2. \begin{align*}log_3 \left (\frac{x^2} {3}\right )\end{align*} = \begin{align*}log_3 x^2 - log_3 3\end{align*} = \begin{align*}2log_3 x - 1\end{align*}

Using the log properties in this way is often referred to as "expanding". In the first example, expanding the log allowed us to simplify, as log2 8 = 3. Similarly, in the second example, we simplified using the log properties, and the fact that log3 3 = 1.

    

 

Examples

Example 1

To condense a log expression, we will use the same properties we used to expand expressions. Consider the expression \begin{align*}log_6 (8) + log_6 (27)\end{align*}. Individually, neither of these expressions has an integer value. The value of log6 8 is between 1 and 2; the value of log6 27 is also between 1 and 2.

However, if we condense the expression, we get:

\begin{align*}log_6 (8) + log_6 (27) = log_6 (8 \cdot 27) = log_6 (216) = 3\end{align*}

Example 2

Expand each expression.

  1. \begin{align*}log_5 25x^2 y\end{align*}

\begin{align*}log_5 (25)x^{2}y = log_5 (25) + log_5 x^{2} + log_5 y = 2 + 2 log_5 x + log_5 y\end{align*}

  1. \begin{align*}log_{10} \left (\frac{100x} {9b}\right )\end{align*}
\begin{align*}log_{10} \left (\frac{100x} {9b}\right )\end{align*} \begin{align*}= log_{10} 100x - log_{10} 9b\end{align*}
\begin{align*}= log_{10} 100 + log_{10} x - \left [log_{10} 9 + log_{10} b\right]\end{align*}
\begin{align*}= 2 + log_{10} x - log_{10} 9 - log_{10} b\end{align*}

Example 3

Condense the expression 2log3 x + log3 5x - log3 (x + 1).

\begin{align*}2log_3 x + log_3 5 x - log_3 (x + 1) = log_3 x_2 + log_3 5x - log_3 (x + 1)\end{align*}

\begin{align*}= log_3 (x^2 (5x)) - log_3 (x + 1)\end{align*}

\begin{align*}=log_3 \left (\frac{5x^3} {x + 1}\right )\end{align*}

Note that not all solutions may be valid, since the argument must be defined. For example, the expression above: \begin{align*}\left (\frac{5x^3} {x + 1}\right )\end{align*} is undefined if x = -1.

Example 4

Condense the expression log2 (x2 - 4) - log2 (x + 2).

\begin{align*}log_2 (x^2 - 4) - log_2 (x + 2)= log_2 \left (\frac{x^2 - 4} {x + 2}\right )\end{align*}

\begin{align*}= log_2 \left (\frac{(x + 2)(x - 2)} {x + 2}\right )\end{align*}

\begin{align*}= log_2 (x - 2)\end{align*}

Note that the argument of a log must be positive. For example, the expressions above are not defined for x ≤ 2 (which allows us to "cancel" (x+2) without worrying about the condition x≠ -2).

Example 5

Condense the following into a single logarithm: \begin{align*}3log_6 x + 2log_6 (3x) - log_6 (2x^3)\end{align*}.

Recall that \begin{align*}3log_x y = log_x y^3\end{align*}

\begin{align*}3log_6 x + 2log_6 (3x) - log_6 (2x^3) \to\end{align*}

\begin{align*}log_6 (x^3 + 3x^2) - log_6 (2x^3) \to\end{align*}

\begin{align*}log_6 (\frac{x^3 + 3x^2}{2x^3}) \to\end{align*}

\begin{align*}log_6 (\frac{x + 3}{2x})\end{align*}

Example 6

Expand the logarithm: \begin{align*}log_2 (\frac{5x^7}{3x^4})\end{align*}.

Reversing a previously used rule gives \begin{align*}log_x (\frac{y}{z}) = log_x y - log_x z\end{align*}.

\begin{align*}log_2 (\frac{5x^7}{3x^4}) \to\end{align*}

\begin{align*}log_2 (\frac{5x^3}{3}) \to\end{align*} (reducing the fraction first)

\begin{align*}log_2 5x^3 - log_2 3\end{align*}

Review

Expand each logarithmic expression:

  1. \begin{align*}log_{5}(ab)\end{align*}
  2. \begin{align*}log_{6}\frac{a}{\sqrt{3}b}\end{align*}
  3. \begin{align*}log_{6}\frac{ab}{c}\end{align*}
  4. If \begin{align*}v = log_x (\frac{4z^2}{y^3})\end{align*} expand \begin{align*}v\end{align*}
  5. \begin{align*}log_2 (\frac{4x^3}{\sqrt{y}})\end{align*}
  6. If \begin{align*}R = log_3 (\frac{2GM}{c^2})\end{align*} expand \begin{align*}R\end{align*}

Condense each logarithmic expression:

  1. \begin{align*}log_{5} A + log_{5} C\end{align*}
  2. \begin{align*}\frac{1}{2}log_{2} C - log_{2} B\end{align*}
  3. \begin{align*}2log_{b} x + 2log_{b} y\end{align*}
  4. \begin{align*}6log_{10}a + log_{10} b\end{align*}
  5. \begin{align*}2log_3 a + 4log_3 b - log_3 c\end{align*}
  6. \begin{align*}\frac{1}{2}log_4 w - 5log_4 z\end{align*}
  7. \begin{align*}(log_{10} x + log_{10} y) - log_{10} w\end{align*}

Simplify:

  1. \begin{align*}log_{10} A^3 - log_{10} B^{\frac{2}{3}} + log_{10} A^{\frac{1}{3}} + log_{10} B^{\frac{5}{3}}\end{align*}
  2. \begin{align*}\frac{log_{9} A^2 - 2log_{9} B}{log_{9} A^2 + log_{9} B^3}\end{align*}
  3. \begin{align*}2ln(AB) - ln(\frac{B}{A})\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.7. 

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Vocabulary

Condensing logs

Condensing logs refers to the process of combining two individual logarithms into a single logarithm.

Expanding logs

Expanding logs refers to the process of splitting a single log into two separate and simpler logs.

log

"log" is the shorthand term for 'the logarithm of', as in "\log_b n" means "the logarithm, base b, of n."

Logarithm

A logarithm is the inverse of an exponential function and is written \log_b a=x such that b^x=a.

Logarithmic

A function or expression is logarithmic if it contains a logarithm.

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