Which equation has $x=4$ as the solution?
$\log _4(3 x+4)=2$
$\log _3(2 x-5)=2$
$\log _x 64=4$
$\log _x 16=4$
Answer (2)
Substitute x = 4 into the first equation lo g 4 ( 3 x + 4 ) = 2 and verify that lo g 4 ( 3 ( 4 ) + 4 ) = lo g 4 ( 16 ) = 2 .
Substitute x = 4 into the second equation lo g 3 ( 2 x − 5 ) = 2 and verify that lo g 3 ( 2 ( 4 ) − 5 ) = lo g 3 ( 3 ) = 1 = 2 .
Substitute x = 4 into the third equation lo g x 64 = 4 and verify that lo g 4 ( 64 ) = 3 = 4 .
Substitute x = 4 into the fourth equation lo g x 16 = 4 and verify that lo g 4 ( 16 ) = 2 = 4 .
Conclude that the first equation has x = 4 as a solution: lo g 4 ( 3 x + 4 ) = 2 .
Explanation
Understanding the Problem We are given four equations and we need to find the equation for which x = 4 is a solution. The equations involve logarithms, so we need to understand how logarithms work. A logarithm is the inverse operation to exponentiation. For example, lo g b a = c means that b c = a . We will substitute x = 4 into each equation and see which one holds true.
Testing Equation 1 Let's test the first equation: lo g 4 ( 3 x + 4 ) = 2 . Substitute x = 4 into the equation: lo g 4 ( 3 ( 4 ) + 4 ) = lo g 4 ( 12 + 4 ) = lo g 4 ( 16 ) Since 4 2 = 16 , we have lo g 4 ( 16 ) = 2 . Thus, the first equation is true when x = 4 .
Testing Equation 2 Now let's test the second equation: lo g 3 ( 2 x − 5 ) = 2 . Substitute x = 4 into the equation: lo g 3 ( 2 ( 4 ) − 5 ) = lo g 3 ( 8 − 5 ) = lo g 3 ( 3 ) Since 3 1 = 3 , we have lo g 3 ( 3 ) = 1 . Since 1 = 2 , the second equation is false when x = 4 .
Testing Equation 3 Next, let's test the third equation: lo g x 64 = 4 . Substitute x = 4 into the equation: lo g 4 ( 64 ) = 4 Since 4 3 = 64 , we have lo g 4 ( 64 ) = 3 . Since 3 = 4 , the third equation is false when x = 4 .
Testing Equation 4 Finally, let's test the fourth equation: lo g x 16 = 4 . Substitute x = 4 into the equation: lo g 4 ( 16 ) = 4 Since 4 2 = 16 , we have lo g 4 ( 16 ) = 2 . Since 2 = 4 , the fourth equation is false when x = 4 .
Conclusion Therefore, only the first equation, lo g 4 ( 3 x + 4 ) = 2 , is true when x = 4 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), measuring the loudness of sound (decibels), and in chemistry to measure the acidity or alkalinity of a substance (pH). Understanding logarithms helps us to quantify and compare these phenomena. For example, an earthquake of magnitude 6 on the Richter scale is ten times stronger than an earthquake of magnitude 5.
The equation that has x = 4 as a solution is lo g 4 ( 3 x + 4 ) = 2 . By substituting x = 4 into this equation, we find it holds true, while the other equations do not. Therefore, the correct choice is the first equation.
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