# Common Logs

## Key Questions

• There are 2 ways.

The math way is to understand how to convert bases:

$\log a = \frac{\ln a}{\ln 10}$

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

Here is an example of $\log 50$:

$= \frac{\ln 50}{\ln} \left(10\right)$
$\approx 1.69897$

See the explanation.

#### Explanation:

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

${\log}_{10} n = {\log}_{b} \frac{n}{\log} _ b 10$ (where $b = e \text{ or } 2$)

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 $\ln 10 \approx 2.302585093$

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 ${\log}_{10} n \approx \ln \frac{n}{2.302585093}$

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