Rewrite the following in the form [tex]$\log (c)$[/tex].
[tex]$3 \log (4)$[/tex]
Answer (2)
Apply the power rule of logarithms: 3 lo g ( 4 ) = lo g ( 4 3 ) .
Calculate 4 3 = 4 × 4 × 4 = 64 .
Substitute the value back into the logarithm: lo g ( 4 3 ) = lo g ( 64 ) .
The expression 3 lo g ( 4 ) is rewritten as lo g ( 64 ) , so the final answer is 64 .
Explanation
Understanding the problem We are given the expression 3 lo g ( 4 ) and we want to rewrite it in the form lo g ( c ) . To do this, we need to use the power rule of logarithms.
Applying the power rule The power rule of logarithms states that a lo g ( b ) = lo g ( b a ) . Applying this rule to our expression, we have:
3 lo g ( 4 ) = lo g ( 4 3 )
Calculating the exponent Now we need to calculate 4 3 . This means 4 × 4 × 4 = 16 × 4 = 64 . Therefore, we have:
lo g ( 4 3 ) = lo g ( 64 )
Final Answer Thus, we have rewritten the expression 3 lo g ( 4 ) in the form lo g ( c ) , where c = 64 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and calculating the loudness of sound in decibels. In finance, logarithms are used to calculate the time it takes for an investment to double at a given interest rate. For example, if you want to know how long it will take for an investment to grow to a certain value, you can use logarithms to solve for the time variable in the compound interest formula. Understanding logarithms helps in making informed decisions in various fields.
The expression 3 lo g ( 4 ) can be rewritten as lo g ( 64 ) using the power rule of logarithms. By calculating 4 3 , which equals 64 , we substitute this value into the logarithm. Thus, 3 lo g ( 4 ) = lo g ( 64 ) .
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