Question

Use logarithm properties to expand the expression ln((x³)z⁵/(y⁷)

Answer 1

To expand the expression ln((x³)z⁵/(y⁷)), using logarithm properties, we find that it can be written as 3*ln(x) + 5*ln(z) - 7*ln(y) by applying properties for logarithms of quotients, products, and exponents.

Step-by-step explanation:

To use logarithm properties to expand the expression ln((x³)z⁵/(y⁷)), we apply several properties of logarithms. First, we utilize the property that the logarithm of a quotient is equal to the difference of the logarithms:

ln(a/b) = ln(a) - ln(b).

Using this, we can separate the numerator and the denominator as follows:

ln((x³)z⁵) - ln(y⁷).

We then apply the property that the logarithm of a product equals the sum of the logarithms:

ln(xy) = ln(x) + ln(y).

This allows us to write:

ln(x³) + ln(z⁵) - ln(y⁷).

Lastly, we apply the property that the logarithm of an exponential expression equals the exponent times the logarithm of the base:

ln(x⁴) = n * ln(x).

This gives us our final expanded expression:

3 * ln(x) + 5 * ln(z) - 7 * ln(y).