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Logarithms – properties ( Radicals) – pg.7/10

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

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

6.  Logarithms of Radicals

 

The logarithm of a radical expression is equal with the radical’s index multiplied with the radicand  logarithm. (Look to radicals’ terminology here)

 

Let’s consider:

[14] {root{q}{p} = r}

By raising to the power {{}{q}{}} both sides of  equality [14] we have:

[15] {p = {r^q}}

By writing each side of equality [15] in a logarithmic form, with same base {{}{m}{}},  we have:

{{log_m}{p}} =  q{log_m}{r}   ({log_m}{r^q} = q {log_m}{r} see pg.6)  or:

 

[16]  {{log_m}{r} =  {{{log_m}{p}}/q}}

 

By replacing the value of {{}{r}{}} from [14] into [16] we have:

 

[17]

{{log_m}(root{q}{p}){}{}={}{}  {{{log_m}p}/q}{}{}={}{}{1/q}({log_m}p)}

 

When p=m {}{right}{}  {log_m}(root{q}{m}) = {1/q}{log_m}m = {1/q}*1 = {1/q}  or we can write:

 

{}{p=m}{}

{}{right}{}

{{log_m}(root{q}{m}) = {1/q}}

 

When p=m^r  {}{right}{}  {log_m}(root{q}{m^r}) = {1/q}{log_m}(m^r) = {1/q}*r*{log_m}m = {1/q}*r*1 = {r/q}  or we can write:

 

{}{p=m^r}{}

{}{}{}{}{}{right}{}{}{}{}{}

 

{log_m}(root{q}{m^r}) = {r/q}

Example:

{log_3}{root{4}{9}} = {1/4}({log_3}9) = {1/4}{log_3}(3^2) = {1/4}*2 = {2/4} ={1/2}

 

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

 

 

 

 

 

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