Question

How to show thatln s 1/2 ln(s^2 + 4) from 0 to [infinity] is equal to ln (s^2 + 4/s^2right) ?

A. Apply L'Hôpital's Rule
B. Use the Mean Value Theorem
C. Apply the Fundamental Theorem of Calculus
D. Use the properties of logarithms

Answer 1

The expression ln(s) - ½ ln(s^2 + 4) can be simplified using the properties of logarithms, which include the logarithm of a ratio and the exponentiation rule.

Step-by-step explanation:

To show how to simplify the expression ln(s) - ½ ln(s^2 + 4), we should use the properties of logarithms. Specifically, the property that allows the subtraction of two logarithms to be rewritten as the logarithm of a ratio, and the property that allows us to bring the exponent in front of the logarithm as a multiplier. Therefore, by rearranging the terms and using these logarithmic properties, we get:

• ln(s) - ½ ln(s^2 + 4) = ln(s) - ln((s^2 + 4)^{1/2})
• = ln(s) - ln(√s^2 + 2)
• = ln(​rac{s}{√s^2 + 2})
• = ln(​rac{s^2}{s^2 + 4})
  since squaring s and the denominator removes the square root.