Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln (b⁴ √a)

Verified step by step guidance

1. Start with the given expression: \(\ln \left(b^{4\sqrt{a}}\right)\).
2. Recall the logarithm power rule: \(\ln(x^k) = k \ln(x)\). Apply this to rewrite the expression as \$4\sqrt{a} \cdot \ln b$.
3. Express \(\sqrt{a}\) in terms of \(a\): \(\sqrt{a} = a^{1/2}\). So the expression becomes \$4 a^{1/2} \cdot \ln b$.
4. Since \(u = \ln a\) and \(v = \ln b\), rewrite \(a^{1/2}\) using the exponential and logarithm relationship: \(a^{1/2} = e^{(1/2) \ln a} = e^{(1/2) u}\).
5. Substitute back into the expression to get \$4 e^{(1/2) u} \cdot v\(, which is the expression in terms of \)u\( and \)v\( without using the \)\ln$ function.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the power rule ln(x^r) = r ln(x) and the product rule ln(xy) = ln(x) + ln(y). These allow rewriting complex logarithmic expressions into sums and multiples of simpler logarithms.

Exponent and Root Relationships

Roots can be expressed as fractional exponents, for example, the fourth root of a is a^(1/4). Understanding this allows rewriting expressions like b^4√a as b^4 * a^(1/4), facilitating the use of logarithm properties.

Substitution in Logarithmic Expressions

Given u = ln(a) and v = ln(b), substitution replaces ln(a) and ln(b) with u and v respectively. This helps express logarithmic expressions in terms of u and v without explicitly using the ln function.