Enrichment, Extension, Application

Question 1

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Exercises

  1. If \( x \) and \( y \) are integers and \( 2^{x+1}+2^x = 3^{y+2}-3^y \), find the values of \( x \) and \( y \).
  2. Find the value(s) of \( x \) such that \( \log_x(19x-30) = 3 \)
  3. If \( x \) is a real number (\( x \ne 0 \)) and \begin{align*} f(x) &= \frac{10^{\left( 1 + \tfrac{4}{x} \right)}}{1 + 10^{\left( 1 + \tfrac{4}{x} \right)}} \\ g(x) &= \log_{10} \left( \frac{f(x)}{1 - f(x)} \right) \end{align*}
    1. Find the range of \( f(x) \).
    2. Sketch the graph of \( g(x) \), indicating all asymptotes.
  4. Find the sum of the logarithms (base \( 10 \)) of all the divisors of \( 1~000~000 \).
  5. Determine all real numbers \( x \) for which \[ (\log_{10} (x))^{\log_{10}(\log_{10} (x))}= 10~000 \]
  6. If \( a^2+b^2 = 7ab \), with \( a, b \gt 0 \), prove that \[ \log \left( \frac{a+b}{3} \right) = \frac{1}{2} \Big( \log (a) +\log (b) \Big) \]
  7. If \( x \) is a real number, solve the equation \[ 3^{2x}+3^x = 20 \]
  8. Prove there is a positive integer \( n \) such that \( 1995^{1994} \lt 1994^n \lt 1995^{1995} \).
  9. Determine all values of \( k \) for which the equation \[ k(2^x) + 2^{-x} = 3 \] has a single root.
  10. If \( (\log_2(x)), (1+\log_4(x)), \) and \( (\log_8 (4x)) \) are consecutive terms of a geometric sequence, determine all possible values of \( x \).
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