Which of the following is equivalent to $\log _7(a) \cdot \log _b(7) ?$
Choose 1 answer:
(A) $\log (7)$
(B) $\log (7 a)$
(C) $\log _a(b)$
(D) $\log _b(a)$
Answer (1)
Use the change of base formula to rewrite lo g b ( 7 ) as l o g 7 ( b ) l o g 7 ( 7 ) .
Simplify the expression lo g 7 ( a ) ⋅ lo g b ( 7 ) to lo g 7 ( a ) ⋅ l o g 7 ( b ) l o g 7 ( 7 ) .
Since lo g 7 ( 7 ) = 1 , the expression becomes l o g 7 ( b ) l o g 7 ( a ) .
Use the change of base formula in reverse to rewrite l o g 7 ( b ) l o g 7 ( a ) as lo g b ( a ) . The final answer is lo g b ( a ) .
Explanation
Understanding the Problem We are given the expression lo g 7 ( a ) ⋅ lo g b ( 7 ) and asked to find an equivalent expression from the given choices. To do this, we will use the change of base formula for logarithms.
Change of Base Formula The change of base formula for logarithms is: lo g x ( y ) = lo g z ( x ) lo g z ( y ) for any valid base z. We can use this formula to rewrite lo g b ( 7 ) in terms of base 7.
Applying the Formula Applying the change of base formula to lo g b ( 7 ) , we get: lo g b ( 7 ) = lo g 7 ( b ) lo g 7 ( 7 )
Substitution Now, we can substitute this back into the original expression: lo g 7 ( a ) ⋅ lo g b ( 7 ) = lo g 7 ( a ) ⋅ lo g 7 ( b ) lo g 7 ( 7 )
Simplification Since lo g 7 ( 7 ) = 1 , the expression simplifies to: lo g 7 ( a ) ⋅ lo g 7 ( b ) 1 = lo g 7 ( b ) lo g 7 ( a )
Reverse Change of Base Using the change of base formula in reverse, we can rewrite this as: lo g 7 ( b ) lo g 7 ( a ) = lo g b ( a )
Final Answer Therefore, the expression lo g 7 ( a ) ⋅ lo g b ( 7 ) is equivalent to lo g b ( a ) . Comparing this with the answer choices, we see that option (D) matches our simplified expression.
Examples
Logarithms are used in many scientific and engineering fields. For example, in chemistry, the pH scale is a logarithmic scale used to measure the acidity or basicity of a solution. The formula is p H = − lo g 10 [ H + ] , where [ H + ] is the concentration of hydrogen ions. Understanding the change of base formula allows chemists to convert between different logarithmic scales, which is essential for accurate measurements and calculations.
