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Exponential and Logarithm Functions

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


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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