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NSW Y12 Maths Standard 1 NSW Y12 Maths Standard 2 NSW Y12 Maths Advanced NSW Y12 Maths Extension 1 NSW Y12 Maths Extension 2

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Sequences and Series Graphs and Equations Differential Calculus Integration Exponential and Log Function Trig Functions Motion and Rates Series and Finance Descriptive Statistics Bivariant Data Analysis Random Variables

NSW Y12 Maths - Advanced Exponential and Log Function Log Functions

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Log Functions Theory

This subtopic is a revision of the subtopic in the year 11 course. \\ The solutions of exponential and logarithmic equations will be covered.\\ \begin{multicols}{2} \textbf{Example 1}\\ %10643 Solve \({e^{2x}} - 2{e^x} - 8 = 0\)\\ \textbf{Example 1 solution}\\ $\begin{aligned}\text { Let } y=e^{x} \rightarrow y^{2}=e^{2 x} \\ y^{2}-2 y-8&=0 \\ (y-4)(y+2)&=0  \\ y=4,4&=-2  \\ e^{x}&=4\quad  e^{x} \neq-2 \\ \therefore\ x=& \log _{e} 4 \\ & =1.39 \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ %7 (30976) Solve \(\ln(6 - x) = 2\ln x\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} \ln(6 - x) &= \ln x^2\\ 6-x &= x^2 \qquad (\ln x \text{ is invertable})\\ x^2 + x - 6 &= 0\\(x + 3)(x - 2) &= 0\\ \therefore x = -3 &\text{ or } x = 2 \end{aligned}$\\ \(x = 2\) is the only valid solution since \(2\ln(-3)\) is not defined. \end{multicols}

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  • Log Functions - Video - Logs - How to solve equations where x is in the power

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