Which of the following is equivalent to [tex]$\frac{1}{\log _b(4)}$[/tex] ?
Choose 1 answer:
A. [tex]$\log _b(4)$[/tex]
B. [tex]$\log _4(b)$[/tex]
C. [tex]$-\log _b(4)$[/tex]
D. [tex]$-\log _4(b)$[/tex]
Answer (2)
lo g 4 ( b )
Explanation
Understanding the problem We are given the expression l o g b ( 4 ) 1 and asked to find an equivalent expression from the given choices. We need to use properties of logarithms to simplify the expression.
Recalling Logarithm Properties Recall the property of logarithms: lo g a ( b ) = l o g b ( a ) 1 . This property tells us that the logarithm of b with base a is the reciprocal of the logarithm of a with base b .
Applying the Property Applying this property to the given expression, we have l o g b ( 4 ) 1 = lo g 4 ( b ) .
Conclusion Therefore, the expression l o g b ( 4 ) 1 is equivalent to lo g 4 ( b ) .
Examples
Logarithms are used to solve exponential equations, which appear in various fields such as finance (calculating compound interest), physics (measuring radioactive decay), and computer science (analyzing algorithm complexity). The change of base formula allows us to convert logarithms from one base to another, which is useful when calculators only support certain bases (like base 10 or base e). For example, if you want to calculate lo g 2 ( 7 ) on a calculator that only has a base 10 logarithm function, you can use the change of base formula: lo g 2 ( 7 ) = l o g 10 ( 2 ) l o g 10 ( 7 ) .
The expression l o g b ( 4 ) 1 is equivalent to lo g 4 ( b ) . Therefore, the correct choice is B. lo g 4 ( b ) .
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