# ❤︎² Logarithms… How? (mathbff)

MIT grad introduces logs and shows how to evaluate them. More videos with Nancy coming in 2017! To skip ahead: 1) For how to understand and evaluate BASIC LOGS, skip to time 0:52. 2) For how to evaluate weirder logs, including the log of 0, 1, a FRACTION, or a NEGATIVE number, skip to time 6:44. 3) For NATURAL LOGS (LN X), skip to time 11:17. 4) For even weirder logs, including SOLVING for X and using the CHANGE-OF-BASE formula, skip to time 14:56.

1) BASIC LOGS: you can read log notation as “log, base 3, of 9 equals X”. The small (subscript) number is called the base. You can always evaluate a log expression by rearranging it into something called exponential form. Every log expression is connected to an exponential expression. In this example, the log is connected to the exponential form “3 to the X power equals 9”. This means, “3 raised to what power gives you 9?” Since 3 raised to the power of 2 equals 9, the answer for X is 2. This is also the answer for the value of the log expression. The log is always equal to the power (or exponent) in the exponential version, and in this case it equals 2. If you want, you can find the log value in your head just by asking yourself what power you need in order to turn the base number into the middle number (“argument” number). Note: if there is no base number in the log expression (no little subscript number), then the base is 10, since 10 is the default base.

2) WEIRDER LOGS (log of 0, 1, a negative number, or a fraction): you can use the same steps to rearrange log expressions that have a fraction, negative number, 0, or 1 in them. You can still rearrange them to be in exponential form just like you can with the basic logs from earlier. The log of 1 will always be 0, since 0 is the only power that can turn a base into 1. The log of 0 will always be undefined, since no power can turn a base into 0. The log of a negative number is undefined in the real number system, since no real power can turn a positive base into a negative number.

3) NATURAL LOGS (ln x): the natural log is just a special type of log where the base is e (the special math constant e, which is approximately 2.718 if you plug it into your calculator). You can use the same steps for rearranging the log expression into exponential form. Just remember that ln x means log, base e.

4) EVEN WEIRDER LOGS (solving for X, change-of-base formula): even if there is an X variable in the log part of an equation, you can still rearrange the equation into exponential form. This will let you solve for X. Sometimes you might need to use the change-of-base formula to evaluate a log expression. If there is no whole number power you know that works, it may actually be a decimal power that you can find by using the change-of-base formula. For example, you can re-write log, base 2, of 7 as (log 7)/(log 2) and use your calculator to find the decimal number if you need it.

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