Explore

Properties of Logs Solutions

The logarithmic properties are used to expand and contract logarithmic expressions and equations. 
The quotient , product , and power rules will be used along with the foundational properties of logs. 
The goal for expanding a logarithmic expression is to allow for algebraic manipulation and comparison .

Properties of Logs

For all rules of logs, the variables a, b, and c are positive real numbers, n is a real number, and $a \neq 1$ .

Properties of Logs	Logarithm Rule(s)	Related Exponent Rule
Foundational Properties	If $a^{0} = 1,$ then $\log_{a} 1 = 0$ If $a^{n} = a^{n},$ then $\log_{a} (a^{n}) = n$ If $a^{1} = a,$ then $\log_{a} (a) = 1$
Quotient Rule	$\log_{a} \frac{b}{c} = \log_{a} b - \log_{a} c$	$\frac{a^{b}}{a^{c}} = a^{b - c}$
Product Rule	$\log_{a} b c = \log_{a} b + \log_{a} c$	$a^{b} \cdot a^{c} = a^{b + c}$
Power Rule	$\log_{a} b^{n} = n \cdot \log_{a} b$	${(a^{b})}^{c} = a^{b c}$

Rewrite radical expressions as fractional exponents before expanding with the properties of logs.

For this lesson, assume all variables and variable expressions are positive values.

Example 1

Expand. Name the property of logs used.

$\log_{2} 5 x y z$

Product

$\log_{2} 5 + \log_{2} x + \log_{2} y + \log_{2} z$

$\log_{x} (4 x^{2} + 12 x + 9)$

Power

$\log_{x} {(2 x + 3)}^{2} 2 \log_{x} (2 x + 3)$

$\log_{b} \frac{3 x - 7}{2 x + 1}$

Quotient

$\log_{b} (3 x - 7) - (\log_{b} (2 x + 1))$

Note

In the first step of expanding with the quotient rule, keeping the denominator in ( ) will help prevent using the incorrect sign of terms.

Example 2

Expand. Explain by listing the logarithmic property used for each step.

$\log_{c} \frac{3 x \sqrt{y z}}{a^{3} b^{2}}$

Implement

$\log_{c} \frac{3 x {(y z)}^{\frac{1}{2}}}{a^{3} b^{2}}$

$\log_{c} 3 x {(y z)}^{\frac{1}{2}} - (\log_{c} a^{3} b^{2}) \log_{c} 3 x + \log_{c} {(y z)}^{\frac{1}{2}} - (\log_{c} a^{3} + \log_{c} b^{2}) \log_{c} 3 + \log_{c} x + \frac{1}{2} \log_{c} (y z) - (3 \log_{c} a + 2 \log_{c} b) \log_{c} 3 + \log_{c} x + \frac{1}{2} (\log_{c} y + \log_{c} z) - (3 \log_{c} a + 2 \log_{c} b) \log_{c} 3 + \log_{c} x + \frac{1}{2} \log_{c} y + \frac{1}{2} \log_{c} z - 3 \log_{c} a - 2 \log_{c} b$

Explain

Radical to fractional exponent
 Quotient rule 
Product 
Power
 Product