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Logarithms – properties (Changing logarithms from one base to another) – pg.8/10

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

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

7. Changing logarithms from one base to another

 

For this demonstration we are going to use the notations in [1] and [2], pg.2.

[18]    {m^n=p}   (shown in [1], pg.2)

[19]    {log_m} p = n   (shown in [2], pg.2)

Let’s assume that:

*  [20]    {p = w^{q}}   or as a logarithmic expression: 

    [21]    {{log_w} p = q}

*  [22]    {m = w^{r}}  or as a logarithmic expression: 

    [23]    {{log_w} m = r}

 

By replacing [20] and [22] into [18] we have:

{{(w^r)}^n = {w^q}}  (see  [13] in § 12 Raising to the power an exponential expression in Exponents (Powers))

[24]    {{w^{r n} } = {w^q}}

From [24] we have {r n = q}  and

[25]    {n= q/r }

By replacing in [25]:

*  {{}{n}{}} from [19] 

*  {{}{q}{}} from [21]

*  {{}{r}{}} from [23]

the expression [25] will become::

[26]  {{log_m}{p} = {{{log_w} p}/{{log_w} m}}}

If  w = 10 (common logarithm), our expression becomes:

[27]  {log_m}{p} = {{{log} p}/{{log} m}}

If  w = e (natural logarithm), our expression becomes:

[28]  {{log_m}{p} = {{ln} p}/{{ln} m}}

 

The general expression for changing a logarithm from one base to another is:

[29]

{{log_m}{p} = {{{log_w} p}/{{log_w} m}}}

The general expression for changing a logarithm from one base to common logarithm is:

{{log_m}{p} =  {{{log}p}/{{log}m}}}

 

The general expression for changing a logarithm from one base to natural logarithm is:

{{log_m}{p} = {{{ln}p}/{{ln}m}}}

Example:

Let’s consider the logarithmic expression:  {log_3} 27 = {log_3}(3^3) = 3

Let’s change this logarithm into logarithms that have different base:

1. Let’s consider the change into a logarithm of base {{}{2}{}}:

 {log_3}{27} = {{{log_2}{27}}/{{log_2}{3}}} = {{{log_2}(3^3)}/{{log_2}{3}}} ={{3{{log_2}{3}}}/{{log_2}{3}}} = 3

 

 2. Let’s consider the change into a logarithm of base {{}{10}{}}:

 {log_3}{27} = {{log{27}}/{log{3}}} = {{log(3^3)}/{log{3}}} ={{3{log{3}}}/{log{3}}} = 3

 

 3.  Let’s consider the change into a logarithm of base {{}{e}{}}:

 {log_3}{27} = {{ln{27}}/{ln{3}}} = {{ln(3^3)}/{ln{3}}} ={{3{ln{3}}}/{ln{3}}} = 3

 

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

 

 

 

 

 

 

 

 

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