2 Simplifying expressions involving logarithms

To simplify an expression involving logarithms their laws, given in Key Point 8, need to be used.

Example 4

Simplify: ${log}_{10}2-{log}_{10}4+{log}_{10}\left(4^2\right)+{log}_{10}\left(\frac{10}{4}\right)$

Solution

The third term ${log}_{10}\left(4^2\right)$ simplifies to $2{log}_{10}4$ and the last term

${log}_{10}\left(\frac{10}{4}\right)={log}_{10}10-{log}_{10}4=1-{log}_{10}4$

So ${log}_{10}2-{log}_{10}4+{log}_{10}\left(4^2\right)+{log}_{10}\left(\frac{10}{4}\right)={log}_{10}2-{log}_{10}4+2{log}_{10}4+1-{log}_{10}4={log}_{10}2+1$

Task!

Simplify the expression:

${log}_{10}\left(\frac{1}{10}\right)-{log}_{10}\left(\frac{10}{27}\right)-{log}_{10}1000$

1. First simplify ${log}_{10}\left(\frac{1}{10}\right)$ :

   Answer

   ${log}_{10}\left(\frac{1}{10}\right)={log}_{10}1-{log}_{10}10=0-1=-1$

2. Now simplify ${log}_{10}\left(\frac{10}{27}\right)$ :

   Answer

   ${log}_{10}\left(\frac{10}{27}\right)={log}_{10}10-{log}_{10}27=1-{log}_{10}27$

3. Now simplify ${log}_{10}1000$ :

   Answer

   3

4. Finally collect all the terms together from (a), (b), (c) and simplify:

   Answer

   $-1-\left(1-{log}_{10}27\right)+3=1+{log}_{10}27$