An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Simplify the expression inside the logarithm using exponent rules: M M − 3 = M − 2 .
Apply the power rule of logarithms: lo g ( M − 2 ) = − 2 lo g ( M ) .
The expanded form of the given logarithmic expression is − 2 lo g ( M ) .
The correct simplification of lo g ( M M − 3 ) is − 2 lo g ( M ) .
Explanation
Understanding the problem We are asked to expand the logarithmic expression lo g ( M M − 3 ) as much as possible and evaluate it without a calculator. We will use properties of exponents and logarithms to simplify the expression.
Simplifying the expression inside the logarithm First, we simplify the expression inside the logarithm using the properties of exponents. Recall that M a M b = M a + b . Therefore, we have M M − 3 = M 1 M − 3 = M 1 + ( − 3 ) = M − 2 .
Applying the power rule of logarithms Now we have lo g ( M − 2 ) . We can use the power rule of logarithms, which states that lo g ( A B ) = B lo g ( A ) . Applying this rule, we get lo g ( M − 2 ) = − 2 lo g ( M ) .
Final Answer Therefore, the expanded form of the given logarithmic expression is − 2 lo g ( M ) . The question asks for the expression to be in the form lo g ( M − 3 ) = □ . However, we found that lo g ( M M − 3 ) = − 2 lo g ( M ) . Thus, the provided form lo g ( M − 3 ) is incorrect based on our calculations.
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale) or the loudness of sound (decibels). They also appear in various scientific and engineering contexts, such as calculating exponential growth or decay, and in computer science for analyzing algorithms. Understanding how to manipulate logarithmic expressions is essential for solving problems in these fields.
The total charge delivered by an electric device with a current of 15.0 A for 30 seconds is 450 C. This corresponds to approximately 2.81 x 10^21 electrons flowing through it. To find the number of electrons, we use the charge of an electron which is about 1.6 x 10^-19 C.
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