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| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| math:log_exp [2020/12/14 02:14] – Added inverse property epix | math:log_exp [2022/01/24 15:36] (current) – fixed missing base epix | ||
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| This property applies to equations that look like $a^x = a^y$. If $x = y$, then you can drop the exponent. The same works for logs (i.e. $\ln x = \ln y$ -> $x = y$). | This property applies to equations that look like $a^x = a^y$. If $x = y$, then you can drop the exponent. The same works for logs (i.e. $\ln x = \ln y$ -> $x = y$). | ||
| ==== Inverse Property ==== | ==== Inverse Property ==== | ||
| + | This one is pretty self explanatory. | ||
| * $a^{\log_{a}x} = x$ | * $a^{\log_{a}x} = x$ | ||
| * $\log_{a}a^x = x$ | * $\log_{a}a^x = x$ | ||
| + | ==== Multiplication Property ==== | ||
| + | The log of a product is the sum of the logs. | ||
| + | * $\log_{a}xy = \log_{a}x + \log_{a}y$ | ||
| + | |||
| + | ==== Division Property ==== | ||
| + | The log of a quotient is the difference of the logs. | ||
| + | * $\log_{a}\frac{x}{y} = \log_{a}x - \log_{a}y$ | ||
| + | |||
| + | ==== Log to a Power Property ==== | ||
| + | The exponent on the argument is the coefficient of the log. | ||
| + | * $\log_{a}x^r = r\log_{a}x$ | ||
| + | |||
| + | ===== Change of Base Formula ===== | ||
| + | While the TI-84 Plus CE has a $\log_{base}$ function((math -> A: logBASE()) that exempts you from needing this for most problems, it is still good to know and even required if you use an older, shittier, calculator like the normal TI-84 Plus. | ||
| + | The formula goes as follows: $\log_{c}a = \frac{\log_{b}a}{\log_{b}c}$ where $a$ is the result, $b$ is the new base, and $c$ is the old base. As a practical example, with the TI-84, the only $\log$ function available is the common log, or $\log_{10}$(($\log$ without a base specified is base 10)), so to calculate values with the TI-84 of different bases, you have to use the change of base formula to base 10. | ||
| + | |||
| + | ===== Quadratic Exponentials ===== | ||
| + | While these types of problems look daunting((and typically contain $e$ to scare poor highschool precalc students)), they are literally the same as solving a quadratic. Take $e^{2x}-4e^x-5=0$ for example. All you need to do is take out the square with the [[math: | ||
