You're probably also familiar that logarithms can be expressed in any base you'd like, like this:
log_{a} x = ( log_{b} x ) / ( log_{b]} a )
For example, if your calculator has only log base 10, and you want to compute log_{2} 16, you could enter
log_{10} 16 / log_{10} 2
We can put these facts together to good use.
To start with, let's try a simple one: express 2^{i} in the a + bi form. We can express 2^{i} as a power of e by solving this equation:
2^{i} = e^{x}
i ln 2 = x
We've just used the logarithm rule I described above in "reverse." So we've just changed the problem to expressing exp(i ln 2) in a + bi form. Now we can just apply Euler's identity, and we get
exp(i ln 2) = cos(ln 2) + i sin(ln 2).
Thus 2^{i} = cos(ln 2) + i sin(ln 2), as we wished to find.
Now let's try 2^{1 + i}. I'm going to skip all the fanfare and just show the steps.
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling We Value Civility
• Positive and compassionate attitudes
• Patience while debating We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving