Use properties of logarithms to expand each logarithmic expression as much as possible.
[tex]\log _3\left(\frac{\sqrt{f}}{27}\right)[/tex]
Answer (2)
Apply the quotient rule: lo g 3 ( 27 f ) = lo g 3 ( f ) − lo g 3 ( 27 ) .
Rewrite the square root as an exponent: f = f 2 1 .
Apply the power rule: lo g 3 ( f 2 1 ) = 2 1 lo g 3 ( f ) .
Evaluate the constant term: lo g 3 ( 27 ) = 3 .
Combine the terms: 2 1 lo g 3 ( f ) − 3 .
2 1 lo g 3 ( f ) − 3
Explanation
Understanding the Problem We are given the logarithmic expression lo g 3 ( 27 f ) and we want to expand it using properties of logarithms.
Logarithmic Properties We will use the following properties of logarithms:
Quotient Rule: lo g b ( y x ) = lo g b ( x ) − lo g b ( y )
Power Rule: lo g b ( x p ) = p lo g b ( x )
Applying Quotient Rule First, we apply the quotient rule to separate the fraction: lo g 3 ( 27 f ) = lo g 3 ( f ) − lo g 3 ( 27 )
Rewriting Square Root Next, we rewrite the square root as an exponent: f = f 2 1 So we have: lo g 3 ( f ) = lo g 3 ( f 2 1 )
Applying Power Rule Now, we apply the power rule to simplify the first term: lo g 3 ( f 2 1 ) = 2 1 lo g 3 ( f )
Evaluating Logarithm We evaluate the second term: lo g 3 ( 27 ) = lo g 3 ( 3 3 ) = 3
Combining Terms Finally, we combine the simplified terms to get the final expanded expression: lo g 3 ( 27 f ) = 2 1 lo g 3 ( f ) − 3
Final Answer Therefore, the expanded form of the given logarithmic expression is: 2 1 lo g 3 ( f ) − 3
Examples
Logarithms are incredibly useful in many real-world scenarios, especially when dealing with exponential growth or decay. For instance, in finance, logarithms help calculate the time it takes for an investment to double at a certain interest rate. In science, they're used to measure the intensity of earthquakes (the Richter scale) or the acidity of a solution (pH scale). Understanding how to expand and simplify logarithmic expressions allows us to manipulate and solve complex equations in these fields more efficiently. For example, if you're analyzing the decay of a radioactive substance, expanding a logarithmic expression might help you isolate the variable representing time, making it easier to determine the half-life of the substance.
To expand lo g 3 ( 27 f ) , apply the quotient rule to separate the logarithm into two parts. Then use the power rule to rewrite the square root as an exponent and evaluate the logarithm of 27. The final expanded form is 2 1 lo g 3 ( f ) − 3 .
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