Question

Which of the following expressions is equivalent to ln (10x²) for all values of æ, including x<0 . Select all that apply and be careful!

a)2·ln (10x)
b)ln (20x³)-ln (2x⁰)
c)ln (40x²)-ln (4)
d)ln (5x)+ln (2x)
e)2· ln (x)+ln (10)

Answer 1

The expressions equivalent to ln (10x²) are option b) ln (20x³)-ln (2x⁰) and option e) 2 · ln (x)+ln (10), derived from the fundamental properties of logarithms.

Step-by-step explanation:

The expressions equivalent to ln (10x²) are b) ln (20x³)-ln (2x⁰) and e) 2 · ln (x)+ln (10). These are derived from the logarithmic properties stating that the logarithm of a product equals the sum of the logarithms, and that the logarithm of a quotient equals the difference of the logarithms.

Let's consider each option:

• a) 2·ln (10x) is not equivalent because it implies that x must be positive to avoid taking the logarithm of a negative number, which is undefined.
• b) ln (20x³)-ln (2) simplifies to ln (10x²) using the quotient rule of logarithms, which states ln (a/b) = ln (a) - ln (b).
• c) ln (40x²)-ln (4) is not equivalent because it simplifies to ln (10x²), but not for all values of x, similar to option a).
• d) ln (5x)+ln (2x) is not equivalent as it simplifies to ln (10x²) only for positive x.
• e) 2 · ln (x)+ln (10) is equivalent because, using the product rule of logarithms, it simplifies to ln (x²) + ln (10), which is ln (10x²).