Question

Expand the expression: ln(10y^2/x^-1)

a) ln(10y^2) - ln(x)
b) ln(10) + 2ln(y) + ln(x)
c) 2ln(10) + ln(y) - ln(x)
d) ln(10) - 2ln(y) - ln(x)

Answer 1

To expand the expression ln(10y^2/x^-1), we apply properties of logarithms such as logarithm of a product, quotient, and power. After applying these rules, the correct expansion is option b) ln(10) + 2ln(y) + ln(x).

Step-by-step explanation:

The question involves the use of exponential functions and their inverse, the natural logarithm (ln), to expand the given expression ln(10y^2/x^-1). To do this, we can apply the properties of logarithms:

• The logarithm of a product is the sum of the logarithms: ln(ab) = ln(a) + ln(b).
• The logarithm of a quotient is the difference of the logarithms: ln(a/b) = ln(a) - ln(b).
• The logarithm of a power is equal to the exponent times the logarithm of the base: ln(a^x) = x * ln(a).

Using these properties, we can expand the initial expression as follows:

ln(10y^2/x^-1)

= ln(10) + ln(y^2) - ln(x^-1)

= ln(10) + 2*ln(y) - (-1)*ln(x)

= ln(10) + 2*ln(y) + ln(x)

Thus, the correct answer is option b) ln(10) + 2ln(y) + ln(x).