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)

검증된 단계별 안내

Start with the given expression: \(\ln \left(b^{4\sqrt{a}}\right)\).

Recall the logarithm power rule: \(\ln(x^k) = k \ln(x)\). Apply this to rewrite the expression as \(4\sqrt{a} \cdot \ln b\).

Express \(\sqrt{a}\) in terms of \(a\): \(\sqrt{a} = a^{1/2}\). So the expression becomes \(4 a^{1/2} \cdot \ln b\).

Since \(u = \ln a\) and \(v = \ln b\), rewrite \(a^{1/2}\) using the exponential and logarithm relationship: \(a^{1/2} = e^{(1/2) \ln a} = e^{(1/2) u}\).

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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)