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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?

Watch This

Embedded Video:

- James Sousa: The Properties of Logarithms

Guidance

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.

Condensing Expressions (Answer to the concept question in the introduction)

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 A

Expand each expression:

a. \begin{align*}log_5 25x^2 y\end{align*} b. \begin{align*}log_{10} \left (\frac{100x} {9b}\right )\end{align*}

Solution:

a. \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*}
b.
\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 B

Condense the expression:

2log3 x + log3 5x - log3 (x + 1)

Solution:

\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 C

Condense the expression:

log2 (x2 - 4) - log2 (x + 2)

Solution:

\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 in Example 'C' above are not defined for x ≤ 2 (which allows us to "cancel" (x+2) without worrying about the condition x≠ -2).

Vocabulary

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

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

Guided Practice

1) Condense the following expressions into a single logarithm:

\begin{align*}log_2 a + log_2 b + log_2 c\end{align*}

2) Condense the expression into a single logarithm:

\begin{align*}log_4 m + log_4 n - 3log_4 x\end{align*}

3) Condense the following into a single logarithm:

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

4) Expand the logarithm:

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

Answers

1) To condense the logs, apply the rule as explained in the lesson above: \begin{align*}log_x y + log_x z = log_x y \cdot z\end{align*}

\begin{align*}log_2 a + log_2 b + log_2 c \to\end{align*}
\begin{align*}log_2 a \cdot b \cdot c\end{align*}

2) Recall that \begin{align*}log_x y - log_x z = log_x \frac{y}{z}\end{align*}

\begin{align*}log_4 m + log_4 n - log_4 x \to\end{align*}
\begin{align*}log_4 \frac{m \cdot n}{x}\end{align*}

3) 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*}

4) Reversing the rule used in Q 2 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*}

Practice

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*}

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