Common Logs

Key Questions

How to find logs in base 10 on a TI-89?

There are 2 ways.

The math way is to understand how to convert bases:

$log a=(ln a)/(ln 10)$

The second way is to use the "CATALOG" button, "L", scroll down to "log" and press enter.

Here is an example of $log 50$ :

$=(ln 50)/ln(10)$
$~~1.69897$

Amory W. · 1 · Sep 13 2014
How do I find the common logarithm of a number?

Answer:

See the explanation.

Explanation:

If you have technology available for the logarithm in some other base ( $e$ or $2$ ), use

$log_10 n = log_b n / log_b 10$ (where $b = e " or "2$ )

With paper and pencil, I don't know a good series for $log_10 n$ .

Probably the simplest way is to use a series for $ln n$ and either a series or memorization for $ln 10 ~~ 2.302585093$

For $ln n$ , let $x=n-1$ and use:

$ln n = ln (1 + x) = x − x^2/2 + x^3/3- x^4/4+x^5/5- * * *$

After you find $ln n$ , use division to get $log_10 n ~~ ln n / 2.302585093$

Jim H · 2 · Sep 21 2015
What is a common logarithm or common log?

Answer:

The inverse of the function $f(x) = 10^x$

Explanation:

The function:

$f(x) = 10^x$

is a continuous, monotonically increasing function from $(-oo, oo)$ onto $(0, oo)$

graph{10^x [-2.664, 2.338, -2, 12.16]}

Its inverse is the common logarithm:

$f^(-1)(y) = log_10(y)$

which as a result is a continuous, monotonically increasing function from $(0, oo)$ onto $(-oo, oo)$ .

graph{log x [-1, 12.203, -1.3, 1.3]}

Note that the exponential function satisfies:

$10^a * 10^b = 10^(a+b)$

Hence its inverse, the common logarithm satisfies:

$log_10 xy = log_10 x + log_10 y$

George C. · 1 · Aug 5 2018