Which of the following is equivalent to [tex]$\frac{1}{\log _3(m)}$[/tex]?
Choose 1 answer:
A. [tex]$\log _m(3)$[/tex]
B. [tex]$\log _3(m)$[/tex]
C. [tex]$-\log _m(3)$[/tex]
D. [tex]$-\log _3(m)$[/tex]
Answer (1)
The problem requires finding an expression equivalent to l o g 3 ( m ) 1 .
Apply the change of base formula for logarithms: lo g a ( b ) = l o g b ( a ) 1 .
Rewrite the given expression using the change of base formula: l o g 3 ( m ) 1 = lo g m ( 3 ) .
The equivalent expression is lo g m ( 3 ) .
Explanation
Understanding the Problem We are given the expression l o g 3 ( m ) 1 and need to find an equivalent expression from the given choices.
Change of Base Formula The key to solving this problem is using the change of base formula for logarithms. A useful form of this formula states that lo g a ( b ) = l o g b ( a ) 1 . This means that the logarithm of b with base a is the reciprocal of the logarithm of a with base b .
Applying the Formula Applying this formula to our expression, we have l o g 3 ( m ) 1 . Using the change of base formula, we can rewrite this as lo g m ( 3 ) .
Finding the Equivalent Expression Comparing this result with the given options, we see that option (A) lo g m ( 3 ) is the correct answer.
Examples
Logarithms are used in many real-world applications, such as measuring the magnitude of earthquakes on the Richter scale. The formula involves logarithms, and understanding how to manipulate logarithmic expressions, like the change of base formula, is crucial for interpreting and using this scale. For example, if we know the intensity of an earthquake relative to a standard level, we can use logarithms to determine its magnitude. Similarly, in finance, logarithmic scales are used to analyze investment growth and risk. Understanding logarithmic transformations helps in making informed decisions based on data.
