Properties of Exponential and Logarithmic Functions

1.

a⁰=1

2.

a^xa^y = a^x+y

3.

a^x/a^y = a^x-y

4.

a^-x = 1/a^x

5.

(a^x)^y = a^xy

6.

$a^{1/x} = \sqrt[\scriptstyle x]{a}$

7.

(ab)^x = a^xb^x

8.

Logarithms are inverse functions of exponentials:

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

9.

• $\log$ is usually a shorthand notation for $\log_{10}$.¹ I generally prefer to explicitly write down the base, but feel free to use the shorthand if you prefer.
• $\ln$ is called the ``natural logarithm''. $\ln x$ means the same thing as $\log_ex$, where $e=2.71828\ldots$ is Napier's number. This very important logarithm comes up in many equations in chemistry. For our purposes, you mostly need to remember that it's just a logarithm that behaves like all other logarithms.

10.

$\log_a 1=0$

11.

$\log_a(xy) = \log_a x + \log_a y$

12.

$\log_a(x/y) = \log_a x - \log_a y$

13.

$\log_a(1/x) = -\log_a x$

14.

$\log_a(x^y) = y\log_a x$

Footnotes

...$\log_{10}$.¹

One of the reasons that I like to explicitly write down the base is that some people, mostly mathematicians, use $\log$ for the natural logarithm rather than the base-10 logarithm. Writing down the base avoids any possible confusion.