Finally have a little time for a post.

So we know how to solve *x*^{2} = 10 by taking the square root of both side of the equation to get *x* = ±3.162… Note that taking the square root of *x*^{2} undoes or reverses the squaring of *x*.

But what do you do if *x* is in the exponent and not the base?

{2}^{x}\hspace{0.33em}{=}\hspace{0.33em}{10}

\]

You can’t take the *x*th root since you don’t know what *x* is. So what to do? From my last post, you saw that log_{2} 10 means “what is the number that I can use as the exponent of 2 so that the answer is 10”. So in the above equation, if I take the log_{2} of both sides, I get

{\log}_{2}{2}^{x}\hspace{0.33em}{=}\hspace{0.33em}{\log}_{2}{10}

\]

The left side of this equation is doing two inverse operations on the number 2 – raising 2 to a power then taking its log. In other words, the left side can be seen as saying “what is the number that I can use as the exponent of 2 so that the answer is 2^{x} ?”. Well the answer to that question is *x*. So the left side is just *x* and the right side is just a calculation:

{\log}_{2}{2}^{x}\hspace{0.33em}{=}\hspace{0.33em}{\log}_{2}{10}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{x}\hspace{0.33em}{=}\hspace{0.33em}{\log}_{2}{10}\hspace{0.33em}{=}\hspace{0.33em}{3}{.}{32192809488}

\]

Well that’s just dandy! Trouble is, without the internet, how do you find log_{2} 10? I have not seen a calculator with a log_{2} *x* button. As mentioned in my last post, calculators usually have buttons to take logs relative to bases 10 and *e*. Well fortunately, there are lots of properties of logs that can help. The one we can use here is

{\log}_{a}{b}^{x}\hspace{0.33em}{=}\hspace{0.33em}{x}\hspace{0.33em}{\log}_{a}{b}

\]

This means that I can take the log with respect to any base, and the *x* can be removed as the exponent. So for our problem, let’s take the log_{10} (the log *x* key on your calculator) of both sides and see what happens:

{\log}_{10}{2}^{x}\hspace{0.33em}{=}\hspace{0.33em}{\log}_{10}{10}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{x}{\log}_{10}{2}\hspace{0.33em}{=}\hspace{0.33em}{1}

\]

Let’s stop here for a moment before I complete the solution. Why is the right side equal to 1? Log_{10} 10 is saying “what power of 10 equals 10?”. The answer is 1 because 10^{1} = 10. On the left side, I used to log property above to bring the *x* in front of the log. Now log_{10} 2 is just a number. You can use the log *x* key on your calculator to find that log_{10} 2 = 0.3010 to four decimal places. So now the equation becomes

{x}{\log}_{10}{2}\hspace{0.33em}{=}\hspace{0.33em}{1}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{0}{.}{301}{x}\hspace{0.33em}{=}\hspace{0.33em}{1}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{x}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{0.301}\hspace{0.33em}{=}\hspace{0.33em}{3}{.}{3219}

\]

So 2^{3.3219} = 10. In my next post on logs, I’ll do more equation solving using logs.