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.