Condense to a single logarithm: $4 \log _9 11-4 \log _9 7$.
Answer (1)
Apply the power rule of logarithms to get lo g 9 ( 1 1 4 ) − lo g 9 ( 7 4 ) .
Apply the quotient rule of logarithms to combine the terms: lo g 9 ( 7 4 1 1 4 ) .
Simplify the expression to obtain the final answer: lo g 9 ( 7 11 ) 4 .
The condensed expression is lo g 9 ( 7 11 ) 4 .
Explanation
Understanding the Problem We are asked to condense the expression 4 lo g 9 11 − 4 lo g 9 7 to a single logarithm.
Applying the Power Rule First, we use the power rule of logarithms, which states that a lo g b x = lo g b x a . Applying this rule to both terms in the expression, we get:
4 lo g 9 11 = lo g 9 ( 1 1 4 ) and 4 lo g 9 7 = lo g 9 ( 7 4 ) .
So our expression becomes lo g 9 ( 1 1 4 ) − lo g 9 ( 7 4 ) .
Applying the Quotient Rule Next, we use the quotient rule of logarithms, which states that lo g b x − lo g b y = lo g b ( x / y ) . Applying this rule, we combine the two logarithmic terms into a single logarithm:
lo g 9 ( 1 1 4 ) − lo g 9 ( 7 4 ) = lo g 9 ( 7 4 1 1 4 ) = lo g 9 ( 7 11 ) 4 .
Final Answer Therefore, the condensed expression is lo g 9 ( 7 11 ) 4 .
Examples
Logarithms are used in many scientific and engineering applications. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. Similarly, the pH scale, which measures the acidity or alkalinity of a solution, is also a logarithmic scale. Understanding how to condense logarithmic expressions can help in simplifying calculations in these areas.
