Express the given quantity as a single logarithm. [tex]$\ln \left(10+x^2\right)+6 \ln x-\ln (x+2)$[/tex]
Answer (2)
Apply the power rule of logarithms to rewrite 6 ln x as ln ( x 6 ) .
Use the product rule of logarithms to combine ln ( 10 + x 2 ) + ln ( x 6 ) into ln (( 10 + x 2 ) x 6 ) .
Apply the quotient rule of logarithms to combine the remaining terms into a single logarithm.
The final expression is ln ( x + 2 x 6 ( 10 + x 2 ) ) .
Explanation
Understanding the Problem We are given the expression ln ( 10 + x 2 ) + 6 ln x − ln ( x + 2 ) . Our goal is to express this as a single logarithm using the properties of logarithms.
Applying the Power Rule First, we use the power rule of logarithms, which states that a ln b = ln b a . Applying this to the term 6 ln x , we get 6 ln x = ln x 6 . So, our expression becomes ln ( 10 + x 2 ) + ln ( x 6 ) − ln ( x + 2 ) .
Applying the Product Rule Next, we use the product rule of logarithms, which states that ln a + ln b = ln ( ab ) . Applying this to the first two terms, we have ln ( 10 + x 2 ) + ln ( x 6 ) = ln (( 10 + x 2 ) x 6 ) . Our expression now looks like ln (( 10 + x 2 ) x 6 ) − ln ( x + 2 ) .
Applying the Quotient Rule Finally, we use the quotient rule of logarithms, which states that ln a − ln b = ln ( b a ) . Applying this to the remaining terms, we get ln (( 10 + x 2 ) x 6 ) − ln ( x + 2 ) = ln ( x + 2 ( 10 + x 2 ) x 6 ) .
Final Answer Therefore, the expression ln ( 10 + x 2 ) + 6 ln x − ln ( x + 2 ) can be expressed as a single logarithm: ln ( x + 2 x 6 ( 10 + x 2 ) ) .
Examples
Logarithms are incredibly useful in many fields, including finance. For example, when calculating compound interest, logarithms can help simplify complex calculations and determine the time it takes for an investment to reach a certain value. Understanding how to combine and simplify logarithmic expressions can make these financial calculations much easier to manage. For instance, if you're comparing different investment options with varying interest rates and compounding periods, expressing the growth as a single logarithm can provide a clear and concise way to evaluate and compare the potential returns.
The expression ln ( 10 + x 2 ) + 6 ln x − ln ( x + 2 ) can be simplified to a single logarithm using logarithmic properties. By applying the power rule, product rule, and quotient rule, we arrive at ln ( x + 2 x 6 ( 10 + x 2 ) ) . This combines all the terms into one logarithmic expression efficiently.
;
