Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible [tex]$\ln \left(\frac{e^4}{2}\right)$[/tex]
Answer (1)
Apply the quotient rule: ln ( 2 e 4 ) = ln ( e 4 ) − ln ( 2 ) .
Apply the power rule: ln ( e 4 ) = 4 ln ( e ) .
Simplify using ln ( e ) = 1 : 4 ln ( e ) = 4 .
The final expanded expression is 4 − ln ( 2 ) .
Explanation
Understanding the Problem We are given the expression ln ( 2 e 4 ) and we want to expand it using properties of logarithms and evaluate without a calculator.
Applying the Quotient Rule We will use the quotient rule for logarithms, which states that ln ( b a ) = ln ( a ) − ln ( b ) . Applying this rule, we get ln ( 2 e 4 ) = ln ( e 4 ) − ln ( 2 )
Applying the Power Rule Next, we use the power rule for logarithms, which states that ln ( a b ) = b ln ( a ) . Applying this rule, we get ln ( e 4 ) = 4 ln ( e )
Simplifying the Expression Since ln ( e ) = 1 , we can simplify the expression further: 4 ln ( e ) = 4 ( 1 ) = 4
Final Answer Substituting this back into our expression, we have ln ( 2 e 4 ) = 4 − ln ( 2 ) Since we are asked to evaluate the expression as much as possible without a calculator, and ln ( 2 ) is a known irrational number, the final answer is 4 − ln ( 2 ) .
Examples
Logarithmic scales are used to represent large ranges of values in a more manageable way. For example, the Richter scale used to measure the magnitude of earthquakes is logarithmic. An earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. Similarly, in chemistry, pH is a logarithmic scale used to measure the acidity or alkalinity of a solution. Understanding logarithmic properties helps in interpreting these scales and making comparisons.
