Which expression is equivalent to $\log _3 \frac{c}{9} ?$
$\log _3 c+\log _3(9)$
$\log _3(9)+\log _3(c)$
$\log _3 c-\log _3(9)$
$\log _3(9)-\log _3(c)$
Answer (1)
Apply the quotient rule of logarithms: lo g 3 9 c = lo g 3 c − lo g 3 9 .
Simplify lo g 3 9 : Since 9 = 3 2 , lo g 3 9 = 2 .
Substitute the simplified value back into the expression: lo g 3 c − 2 .
The equivalent expression is lo g 3 c − lo g 3 ( 9 ) .
Explanation
Understanding the Problem We are given the expression lo g 3 9 c and asked to find an equivalent expression from the given options. We will use logarithm properties to rewrite the given expression.
Applying the Quotient Rule We will use the quotient rule of logarithms, which states that lo g b y x = lo g b x − lo g b y . Applying this rule to the given expression, we have lo g 3 9 c = lo g 3 c − lo g 3 9.
Simplifying the Logarithm Now we need to simplify lo g 3 9 . Since 9 = 3 2 , we have lo g 3 9 = lo g 3 3 2 . Using the power rule of logarithms, lo g b x p = p lo g b x , we get lo g 3 3 2 = 2 lo g 3 3 . Since lo g b b = 1 , we have lo g 3 3 = 1 , so lo g 3 9 = 2 × 1 = 2 .
Therefore, lo g 3 9 c = lo g 3 c − 2 .
Finding the Equivalent Expression Comparing our result with the given options, we see that the equivalent expression is lo g 3 c − lo g 3 ( 9 ) .
Final Answer Therefore, the expression equivalent to lo g 3 9 c is lo g 3 c − lo g 3 ( 9 ) .
Examples
Logarithms are used to simplify calculations in various fields, such as calculating the magnitude of earthquakes on the Richter scale, measuring the acidity or alkalinity of a solution using pH, and determining the loudness of sound in decibels. The properties of logarithms, like the quotient rule used here, help in simplifying complex expressions and making calculations easier.
