This tutorial covers material encountered in chapter 5 of the VCE Mathematical Methods Textbook, namely:
- Exponential functions
- Index laws
- Log functions
- Log laws and change of base
Q1 – Graphs, Intercepts and Asymptotes of Exponential & Logarithm Functions
Q2 – Inverses of Exponential & Logarithm Functions
Q3 – Exponential & Logarithm Functions
Q4 – Solving Exponential & Logarithm Functions
Q5 – Intercepts of Logarithm Functions
Q6 – Solving Natural Exponential Function
Q7 – Sum and Product of Natural Exponential Functions
Worksheet
Q1. Sketch the graph of each of the following functions, clearly indicating the axis intercepts and any asymptotes (Note that \(\log_e(x)=\ln(x)\)):
(a) \(f(x)=e^x-3\)
(b) \(f(x)=2^{-x}+4\)
(c) \(g(x)=\dfrac{1}{3}(e^x-4)\)
(d) \(g(x)=5-e^{-x}\)
(e) \(h(x)=\ln(3x+2)\)
(f) \(h(x)=-\ln(x-3)\)
(g) \(j(x)=\ln(2-x)\)
Q2. Find \(f^{-1}\) for each of the following functions:
(a) \(f:\R \to (-3,\infty),\, f(x)=e^{2x}-3\)
(b) \(f:(3,\infty) \to \R,\,f(x)=4\ln(x-3)\)
(c) \(f:(-\frac{1}{2},\infty) \to \R,\,f(x)=\log_{10}(2x+1)\)
(d) \(f:\R \to (3,\infty),\, f(x)=2^x+3\)
Q3. For each of the following functions, find \(y\) in terms of \(x\):
(a) \(\ln(y)=\ln(2x+5)\)
(b) \(\log_2(2y)=3\log_2(3x+1)\)
(c) \(\log_{10}(y)=-3+4\log_{10}(x)\)
(d) \(\ln(y)=2x+3\)
Q4. Solve each equation for \(x\), expressing any logarithms in the answer with base \(e\):
(a) \(3^{2x}-3^{x}-3=0\)
(b) \(log_{10}(3x)-2=0\)
(c) \(5^{2x}-2(5^{x})-3=0\)
Q5. The graph of \(f(x)=2\log_2(2x+1)-5\) intersects the axes at the the points \((a, 0)\) and \((0, b)\) and passes through the point \((c, 3)\) for \(a,b,c \in \R\). Find \(a, b\) and \(c\).
Q6. Solve for \(x\) if \(4e^{2x}=251\).
Q7. Bonus question: Let \(f(x)=\dfrac{1}{2}(e^x+e^{-x})\) and \(g(x)=\dfrac{1}{2}(e^x-e^{-x})\):
(a) Show that \(f\) is an even function
(b) Find \(f(u)+f(-u)\)
(c) Find \(f(u)-f(-u)\)
(d) Show that \(g\) is an odd function
(e) Find \(f(x)+g(x),\, f(x)-g(x),\, f(x)g(x)\)
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