The 7 Most Important Equations for Your Retirement: The Fascinating People and Ideas Behind Planning Your Retirement Income

APPENDIX

CRASH COURSE ON NATURAL AND UNNATURAL LOGARITHMS

I'm sure it has been many years (decades?) since you were in high school, so allow me to refresh your memory about the natural logarithm function and its mirror image, the natural exponent function. Both functions—or buttons on your calculator, if you like—feature prominently in the most important equations of this book. Natural logarithms are actually close cousins to common (or base 10) logarithms. I'll explain the connection in a moment.

First, the notation I use for the natural logarithm of any number n—for example, the number 15, 2 or 0—is the expression ln[n]. That is the letter l, the letter n and then square brackets containing the number you would like to “lon” (which is how it is pronounced). In parallel, the notation I use for the natural exponent of any number n is the expression eⁿ. Any good business or scientific calculator has a function button that converts numbers into their natural logarithm and/or can then reverse the process to recover the original number. Okay, now let me explain what the buttons do.

You might remember your common (base 10) logarithm, which is based on the same idea. The common logarithm (pronounced “log”) is another button on your calculator. One is LN and the other is LOG. The common logarithm of any number x is equal to the value—call it y—such that 10 to the power of that y recovers ...