Which of the following is equivalent to [tex]$\log _3(b) \cdot \log _b(27)$[/tex]?
A. 3
B. 9
C. [tex]$\log (9)$[/tex]
D. [tex]$\log _b(3)$[/tex]
Answer (2)
Use the change of base formula to rewrite lo g b ( 27 ) as l o g 3 ( b ) l o g 3 ( 27 ) .
Substitute this into the original expression: lo g 3 ( b ) ⋅ l o g 3 ( b ) l o g 3 ( 27 ) .
Simplify the expression by cancelling out lo g 3 ( b ) .
Evaluate lo g 3 ( 27 ) to get 3. The final answer is 3 .
Explanation
Understanding the Problem We are given the expression lo g 3 ( b ) ⋅ lo g b ( 27 ) and asked to find an equivalent expression from the given choices.
Applying Change of Base Formula We can use the change of base formula to rewrite the second term. The change of base formula states that lo g a ( x ) = l o g c ( a ) l o g c ( x ) . Applying this to lo g b ( 27 ) , we can change the base to 3, so we have lo g b ( 27 ) = l o g 3 ( b ) l o g 3 ( 27 ) .
Substitution Now, substitute this back into the original expression: lo g 3 ( b ) ⋅ lo g b ( 27 ) = lo g 3 ( b ) ⋅ l o g 3 ( b ) l o g 3 ( 27 ) .
Simplification We can cancel out the lo g 3 ( b ) terms in the numerator and the denominator, which simplifies the expression to lo g 3 ( 27 ) .
Evaluating the Logarithm Since 3 3 = 27 , we have lo g 3 ( 27 ) = 3 . Therefore, the expression simplifies to 3.
Final Answer Thus, the expression lo g 3 ( b ) ⋅ lo g b ( 27 ) is equivalent to 3.
Examples
Logarithms are used to solve many problems related to exponential growth and decay. For example, calculating the time it takes for an investment to double at a certain interest rate involves logarithms. The change of base formula allows us to convert logarithms from one base to another, which is useful when calculators only have common logarithms (base 10) or natural logarithms (base e). This problem demonstrates a simple application of the change of base formula and logarithmic properties.
By applying the change of base formula and simplifying, we find that lo g 3 ( b ) ⋅ lo g b ( 27 ) is equivalent to 3. Therefore, the correct choice is option A: 3.
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