Application of laws of logarithm

The laws of logarithm are used to solve complicated questions.

These are the four laws of logarithm

• $\log <em>a{(mn)}=\log </em>am+\log _an$

• $\log <em>a{(\frac{m}{n})}=\log </em>am-\log _an$

• $\log <em>a{(m^n)}=n\log </em>am$

• $\log <em>a{(n)}=\frac{\log </em>bn}{\log _ba}$

Example#1

Find the value of $\frac{2391\times30.72}{23.34}$ by using logarithm.

Solution:

• First let the number equal to a variable Let; $x=\frac{2391\times30.72}{23.34}$

• Take log on both side

  $\log x=\log(\frac{2391\times30.72}{23.34})$

• Simplify divisions if there are any by using the logarithm law

                                       $\\therefore \\log_a(\\frac{x}{y})=\\log_a x -\\log_a y$
  

  $\log x=\log({2391\times30.72})-\log({23.34})$

• Simplify multiplications if there are any by using the logarithm law

                                      $\\therefore \\log_a(xy)=\\log_a(x)+\\log_a(y)$
  

  $\log x=\log(2391)+\log (30.72)-\log({23.34})$

• Use the log table to find the value of logarithm of numbers

  By referring to log table we have:

                                  $\\therefore \\log(2391)=3.3786$
  
                                  $\\therefore \\log(30.72)=1.4874$
  
                                  $\\therefore \\log(23.34)=1.3681$
  

$\log x=3.3786+1.4874-1.3681$

• Simplify

  $\log x=3.4979$

• Take anti log on both side

  $\text{antilog}( \log x)=\text{antilog}(3.4979)$

• Use antilog table to find the value of anti log of a number

                                   $\\therefore \\text{antilog}( \\log x)=x$
  

  $\text{antilog}(3.4979)\approx 3147.023$

• Put the value

  $x\approx 3147.023$

• Hence the answer of $\frac{2391\times30.72}{23.34}=3147.023$