Logarithms and Exponentials

An exponential is $a^x=n$, a logarithm is $\log_{a}n=x$ where $a$ is the base, $x$ is the power, and $n$ is the result. Exponentials and logarithms are inverses of eachother. The constant $e$ is a special case, in which its inverse is the natural log, or $\ln n=x$, which is equivalent to $log_{e}n=x$.

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 one is pretty self explanatory.

• $a^{\log_{a}x} = x$
• $\log_{a}a^x = x$

While the TI-84 Plus CE has a $\log_{base}$ function¹⁾ 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 $\log_{10}$²⁾, so to calculate values with the TI-84 of different bases, you have to use the change of base formula to base 10.

While these types of problems look daunting³⁾, 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 Exponential Rules. So the equation above would become $(e^x)^2-4e^x-5=0$, and there you see the quadratic more clearly⁴⁾. Then, solve the quadratic as usual.