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Correspondence

What you have learned so far has all involved taking an equation's image and making incremental changes or manipulations to it. There are instances, however, in which you will need to make some wholesale change to the entire equation. A good example is in solving certain equations containing logarithms, such as this one.

$\begin{displaymath}3\log(x+1)=6\end{displaymath}$

(This is a base 10 logarithm.) Using symbol motion would get you here:

$\begin{displaymath}\log(x+1)=2\end{displaymath}$

To go further, you would need to use the property that states that $\log_{b}a=c$ means $b^{c}=a$ . Correspondence is when you construct a new image, based on information such as this, that contains the equation in a different form. In this case, you would first have an image like this,

$\includegraphics{figures/tech/fig_tech_a.eps}$

and substitute it with an image like this,

$\includegraphics{figures/tech/fig_tech_b.eps}$

which you can use normal arithmetic and symbol motion to solve. Another example of this process is when you use the quadratic formula to solve an equation.

$\includegraphics{figures/tech/fig_tech_c.eps}$

You would substitute that image with this one.

$\includegraphics{figures/tech/fig_tech_d.eps}$

This is nothing new - everyone learns to do things like this in their first or second algebra course. What is new is that you are doing this kind of equation-wide substitution mentally. You are applying it to the representation of the equation you have in your mind's eye. It may take practice before you can do it well (though it's also possible it will happen for you immediately). It is a simple process, and by the time you have some experience at doing this, it will be second nature.

Next: Exercises Up: Some Advanced Techniques Previous: Windowing