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\( \left. \begin{array} { l l } { e ^ { 2 \ln ( 2 x - 3 ) } = 3 x } \\ { ( x ^ { 2 } - 5 ) ^ { \ln x } = x } & { 5.9 } \end{array} \right. \)

Ask by Todd Turnbull. in South Africa
Mar 15,2025

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The solutions are: 1. \( x = 3 \) 2. \( x = \sqrt{5 + e} \)

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The Deep Dive

Did you know that logarithms were introduced in the early 17th century by John Napier? They revolutionized calculations, enabling easier multiplication and division by transforming them into addition and subtraction problems through the use of logarithmic tables. This innovation was particularly useful for navigators and astronomers, leading to major advancements in science and exploration! Now, let's talk about those equations you’ve got! For \( e^{2 \ln(2x - 3)} = 3x \), remember that \( e^{\ln(a)} = a \), so simplify it to find \( 2x - 3 = 3x \) and solve for x. As for \( (x^{2} - 5)^{\ln x} = x \), consider breaking it down step-by-step by taking \( \ln \) on both sides for clarity. This approach helps you avoid common oversights!

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