3 Logs to base 10 and natural logs

In practice only two kinds of logarithms are commonly used, those to base 10, written ${log}_{10}$ (or just simply $log$ ) and those to base $\text{e}$ , written ${log}_{\text{e}}$ or more usually $ln$ (called natural logarithms ). Most scientific calculators will determine the logarithm to base 10 and to base $\text{e}$ . For example, $log13=1.11394$ (implying $10^{1.11394}=13$ ),  $ln23=3.13549$ (implying ${\text{e}}^{3.13549}=23$ )

Task!

Use your calculator to determine

1. $log10$ ,
2. $log1000000$ ,
3. $log0.1$

Answer

Each of the above results could be determined directly, without the use of a calculator. For example:

Since ${log}_{a}a=1$ then $log10\left(\equiv {log}_{10}10\right)=1$ .

Since ${log}_a{A}^k=k{log}_{a}A$ then $log1000000=log10^6=6log10=6$ .

Since ${log}_a\left(\frac{A}{B}\right)={log}_{a}A-{log}_{a}B$ and ${log}_{a}1=0$ and ${log}_{a}a=0$ , then

$log0.1=log\left(\frac{1}{10}\right)=log1-log\left(10\right)=-1$

Task!

Use your calculator to determine 

1. $ln29.42$ ,
2. $ln\text{e}$ ,
3. $ln0.1$

Answer