# Enrichment, Extension, Application

1 point

## 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 $$x$$.