\(\ln \frac{2 a}{b}=\)
Answer (1)
ln b 2 a can be rewritten using logarithm properties:
Apply the quotient rule: ln b 2 a = ln ( 2 a ) − ln ( b ) .
Apply the product rule: ln ( 2 a ) = ln ( 2 ) + ln ( a ) .
Combine the results: ln b 2 a = ln ( 2 ) + ln ( a ) − ln ( b ) .
The final expression is: ln ( 2 ) + ln ( a ) − ln ( b ) .
Explanation
Understanding the Problem We are given the expression ln b 2 a and we want to rewrite it using properties of logarithms.
Applying the Quotient Rule We will use the quotient rule of logarithms, which states that ln ( y x ) = ln ( x ) − ln ( y ) . Applying this rule to our expression, we get: ln b 2 a = ln ( 2 a ) − ln ( b ) .
Applying the Product Rule Next, we will use the product rule of logarithms, which states that ln ( x y ) = ln ( x ) + ln ( y ) . Applying this rule to ln ( 2 a ) , we get: ln ( 2 a ) = ln ( 2 ) + ln ( a ) .
Final Expression Substituting this back into our expression, we have: ln b 2 a = ln ( 2 a ) − ln ( b ) = ln ( 2 ) + ln ( a ) − ln ( b ) .
Examples
Logarithms are used to simplify calculations in various fields, such as finance and physics. For example, in finance, the future value of an investment can be calculated using exponential functions, and logarithms can be used to solve for the time it takes for an investment to reach a certain value. In physics, logarithms are used to describe the intensity of sound or the magnitude of earthquakes. Understanding how to manipulate logarithmic expressions is crucial for solving these types of problems.
