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Logarithms – properties (Exponents (Powers)) – pg.6/10

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

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

5.  Logarithms of Exponents (Powers)

 

Logarithm of an exponential expression is equal with the exponent multiply with the base logarithm.

 

We know that:

{p^q = underline {p*p*...}under {q times}}

 

When we change both sides of the above equality into logarithmic expressions with same base {{}{m}{}}, we have:

 

{{{log_m}{(p ^q)}} = {{log_m}{underline {(p*p*..)}under {q times}}} =  {underline {{log_m}{p} + {log_m}{p} + ...} under {q times}} =  {q{log_m}{p}}} (logarithm of products, [8], pg. 3)

 

Or we can write:

 

[13]

{log_m}{(p ^q)} = q{log_m}{p}

 

When p=m, {log_m}(m^q) = q{log_m}m = q*1 = q  or we can write:

 

{}{p=m}{}

{}{right}{}

{}{{log_m}(m^q) = q}{}{}

Example:

{log_2}(8^4) = {log_2}{(2^3)^4} ={log_2}{2^(3*4)} = {log_2}{2^12} =12{log_2}2 =12

OR can be solved with a method below:

{log_2}(8^4)  = 4{log_2}8 = 4{log_2}(2^3) = 4*3 = 12

same result, two different methods of arriving to it

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

 

 

 

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