• Home
• Inner Algebra
• Integrate Your Brain

Correspondence

1. Take each of these equations, and using symbol motion, reduce them to the form $\log_{b}a=c$. Then visualize them in the converted form, $b^{a}=c$, and finish solving the equation from there. (For example, take the equation $2\log x=4$. First you simplify it to $\log x=2$, using symbol motion. Then you visualize it in its converted form, $10^{2}=x$, or x = 100.
   1. $3\log(x+1)=3$
   2. $-2+\ln x=4$
   3. $3+2\log(2x+20)=7$
2. Find the root of each quadratic equation below. (Since we haven't covered how to deal with multiple solutions yet, just find the more positive root, $x=\frac{1}{2a}\left(-b+\sqrt{b^{2}-4ac}\right)$.) Solve it by rearranging the equation to the form $ax^{2}+bx+c=0$, using that to construct the equation for the more positive root above, then solving from there. Do all of this mentally, using symbol motion, correspondence, etc. For example, if the equation is $2x^{2}-3=x$, you will visualize it, use symbol motion to change it to $2x^{2}-x-3=0,$ then convert that to
   
   $$x=\frac{1}{2(2)}\left(-(-1)+\sqrt{(-1)^{2}-4(2)(-3)}\right)=\frac{3}{2}$$
   
   
   Since visualizing that whole expression may be tricky, you can visualize and solve just $\sqrt{b^{2}-4ac}$ first. Then visualizing the rest is easier: $x=\frac{1}{2(2)}\left(-(-1)+5\right)=\frac{3}{2}$.
   1. $x^{2}+2x-3=0$
   2. $-x^{2}-4x=3$
   3. $6x-5=x^{2}$
   4. $-2x^{2}-8x=6$

Solutions: 1(a) x = 9, (b) $x=e^{6}$, (c) x = 40. 2(a) x = 1, (b) x = -3, (c) x = 1, (d) x = -3