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

The inverse of the function $f \left(x\right) = {10}^{x}$

#### Explanation:

The function:

$f \left(x\right) = {10}^{x}$

is a continuous, monotonically increasing function from $\left(- \infty , \infty\right)$ onto $\left(0 , \infty\right)$

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

Its inverse is the common logarithm:

${f}^{- 1} \left(y\right) = {\log}_{10} \left(y\right)$

which as a result is a continuous, monotonically increasing function from $\left(0 , \infty\right)$ onto $\left(- \infty , \infty\right)$.

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

Note that the exponential function satisfies:

${10}^{a} \cdot {10}^{b} = {10}^{a + b}$

Hence its inverse, the common logarithm satisfies:

${\log}_{10} x y = {\log}_{10} x + {\log}_{10} y$