Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln √(a³/b⁵)

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Start with the given expression: \(\ln \sqrt{\frac{a^3}{b^5}}\).

Rewrite the square root as an exponent of 1/2: \(\ln \left( \frac{a^3}{b^5} \right)^{\frac{1}{2}}\).

Use the logarithm power rule: \(\ln \left( x^r \right) = r \ln x\), so this becomes \(\frac{1}{2} \ln \left( \frac{a^3}{b^5} \right)\).

Apply the logarithm quotient rule: \(\ln \left( \frac{x}{y} \right) = \ln x - \ln y\), so rewrite as \(\frac{1}{2} ( \ln a^3 - \ln b^5 )\).

Use the logarithm power rule again on each term: \(\frac{1}{2} ( 3 \ln a - 5 \ln b )\), then substitute \(\ln a = u\) and \(\ln b = v\) to get \(\frac{1}{2} (3u - 5v)\).

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Properties of Logarithms

Logarithmic properties allow the simplification of expressions involving logs. Key rules include the product rule (ln(xy) = ln x + ln y), the quotient rule (ln(x/y) = ln x - ln y), and the power rule (ln(x^r) = r ln x). These properties help rewrite complex logarithmic expressions in simpler forms.

Natural Logarithm and Its Inverse

The natural logarithm (ln) is the inverse of the exponential function with base e. Understanding that ln a = u means a = e^u helps in expressing variables in terms of u and v. This relationship is crucial for rewriting expressions without the ln function.

Exponent Rules and Radicals

Exponent rules govern how powers and roots are manipulated, such as √(x) = x^(1/2) and (x^m)^n = x^(mn). Applying these rules allows the expression inside the logarithm to be rewritten as a single power, facilitating substitution using u and v without 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 √(a3/b5)