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Logarithm – properties (logarithms of a division between two numbers and the difference of two logarithms with same base) – pg.5/10

Posted in Math Theory by Lia on 04/08/2013

pg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

4. Logarithm of a division between two numbers and the difference of two logarithms with same base.

 

The logarithm of a division between two numbers is equal with the difference of the numbers’ logarithmic expression.

The difference of two logarithmic expressions is the reverse to the logarithms of a division.

 

If we divide the exponential expressions in [4], pg.3, we have:  {{p/q} = {{m^n}/{m^r}}} or we can write:

[10]  {{p/q} = {m^{n - r}}} (see Exponents (Powers) §3)

By transforming the exponential expression [10] into a logarithmic expression, we have:

       {{n - r} = {log_m}(p/q)}

 From [3], pg.3, we have: {{n - r} = {log_m}p - {log_m}q}

 

This means that:

[11]   (logarithm of division)

{log_m}(p/q) = {log_m}p - {log_m}q

 

OR

[12]   (difference of logarithms)

{log_m}p - {log_m}q = {log_m}(p/q)

 

Example:

  • (division)

{log_3}(27/9) = {log_3}27 - {log_3}9 = {log_3}(3^3) - {log_3}(3^2) = 3 - 2 = 1  

  • (subtraction)

 {log_3}27 - {log_3}9 = {log_3}(27/9) = {log_3}3  = 1  

 

pg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

 

 

 

 

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