An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Logarithmic functions are only defined for positive x values.
Table 2 has negative x values, so it cannot represent a logarithmic function.
Table 1 has positive x values, so it could represent a logarithmic function.
Testing the points in Table 1, we find that they all satisfy the equation y = lo g 2 x .
Therefore, Table 1 represents a logarithmic function in the form y = lo g b x when 1"> b > 1 . The answer is Table 1. T ab l e 1
Explanation
Analyzing the Problem We are given two tables of x and y values and asked to identify which table represents a logarithmic function of the form $y =
\log_b x w h ere b > 1$.
Table 1 contains the following (x, y) pairs: ( 8 1 , − 3 ) , ( 4 1 , − 2 ) , ( 2 1 , − 1 ) , ( 1 , 0 ) , ( 2 , 1 ) .
Table 2 contains the following (x, y) pairs: ( − 1.9 , − 2.096 ) , ( − 1.75 , − 1.262 ) .
The base of the logarithm, b , is greater than 1.
Logarithmic functions are only defined for positive values of x.
Checking the Domain First, let's check the domain of logarithmic functions. Logarithmic functions, such as y = lo g b x , are only defined for 0"> x > 0 . This means that the x-values must be positive. Table 2 contains negative x-values, so it cannot represent a logarithmic function.
Analyzing Table 1 Now, let's examine Table 1. The x-values are 8 1 , 4 1 , 2 1 , 1 , 2 , which are all positive. So, Table 1 could represent a logarithmic function. We need to find a base 1"> b > 1 such that y = lo g b x for all pairs in the table.
Finding the Base Let's test the first point ( 8 1 , − 3 ) . If this point lies on the curve y = lo g b x , then − 3 = lo g b 8 1 . This means b − 3 = 8 1 . Taking the reciprocal of both sides, we get b 3 = 8 . Taking the cube root of both sides, we find b = 2 .
Verifying the Base Now we need to check if the other points in Table 1 also satisfy y = lo g 2 x .
For the point ( 4 1 , − 2 ) , we have − 2 = lo g 2 4 1 , which means 2 − 2 = 4 1 . This is true.
For the point ( 2 1 , − 1 ) , we have − 1 = lo g 2 2 1 , which means 2 − 1 = 2 1 . This is true.
For the point ( 1 , 0 ) , we have 0 = lo g 2 1 , which means 2 0 = 1 . This is true.
For the point ( 2 , 1 ) , we have 1 = lo g 2 2 , which means 2 1 = 2 . This is true.
Conclusion Since all the points in Table 1 satisfy the equation y = lo g 2 x and 1"> 2 > 1 , Table 1 represents a logarithmic function of the form y = lo g b x with 1"> b > 1 .
Examples
Logarithmic functions are incredibly useful in many real-world scenarios. For example, the Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. A quake of magnitude 6 is ten times more powerful than a quake of magnitude 5. Similarly, the decibel scale, used to measure sound intensity, is also logarithmic. These scales allow us to represent a wide range of values in a more manageable way. Understanding logarithmic functions helps us interpret and analyze data in fields like seismology, acoustics, and finance, where exponential growth and decay are common.
The device delivers a charge of 450 C in 30 seco n d s at a current of 15.0 A . This corresponds to approximately 2.81 × 1 0 21 electrons flowing through it. The calculation involves using the definitions of current and electric charge.
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