Question

Express the given quantity as a single logarithm: 15ln(x²)⁵ - 12ln(x) - ln((x²)(3x²))²?

Answer 1

To express the given quantity as a single logarithm, apply the logarithmic properties. Start by rewriting the given expression using the properties of logarithms, then simplify the expression by combining like terms. Finally, express the resulting expression as a single logarithm.

Step-by-step explanation:

To express the given quantity as a single logarithm, we'll apply the logarithmic properties.

1. The logarithm of a product of two numbers is the sum of the logarithms of the two numbers. So, 15ln(x²)⁵ - 12ln(x) can be rewritten as ln((x²)⁵)⁻¹ - ln(x)¹².
2. The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. So, ln((x²)⁵)⁻¹ - ln(x)¹² becomes -5ln(x²) - 12ln(x).
3. Using another logarithmic property, the resulting expression can be written as a single logarithm: ln(x⁻¹⁰(x²)⁻¹²).