Respuesta :

The simplified expression of  [tex]log_8 (\frac{64r^{6}}{m^{3}})[/tex] is [tex]2 +6\; log_8 \;r - 3 \;log_8\; m[/tex]

[tex]log_8 (\frac{64r^{6}}{m^{3}})=2 +6\; log_8 \;r - 3 \;log_8\; m[/tex]

What are Logarithm and its rules?

A logarithm is the power to which a number must be raised in order to get some other number .

Rules:

  • Product:                 log(xy)=log(x)+log(y)
  • Quotient:                log(x/y)=log(x)−log(y)
  • Log of power: log [tex]x^{y}[/tex] = y log(x)
  • Log of e:                  log(e)=1
  • Log of one:              log(1)=0

The given expression is: [tex]log_8 (\frac{64r^{6}}{m^{3}})[/tex]

Using the property,

[tex]log(\frac{a}{b})[/tex] = log a - log b

So,

[tex]log_8 (\frac{64r^{6}}{m^{3}})= log_8 \;64r^{6} - log_8\;{m^{3}}[/tex]

Now using the property,

[tex]log a^{b} = b \;log a[/tex]

So, [tex]log_8 (\frac{64r^{6}}{m^{3}})= log_8 \;(2r)^{6} - log_8\;{m^{3}}[/tex]

     [tex]log_8 (\frac{64r^{6}}{m^{3}})= 6\;log_8 (2r) - 3 \;log_8 m[/tex]

Now using the property,

log (ab)= log a + log b

we get, [tex]log_8 (\frac{64r^{6}}{m^{3}})= 6 [\;log_8 \;2 + log_8 \;r ]- 3 \;log_8 m[/tex]

[tex]log_8 (\frac{64r^{6}}{m^{3}})= 6 \;log_8 \;2 +6\; log_8 \;r - 3 \;log_8\; m[/tex]

[tex]log_8 (\frac{64r^{6}}{m^{3}})= 6 * \frac{1}{3} +6\; log_8 \;r - 3 \;log_8\; m[/tex]

[tex]log_8 (\frac{64r^{6}}{m^{3}})=2 +6\; log_8 \;r - 3 \;log_8\; m[/tex]

Hence the simplified expression is :[tex]log_8 (\frac{64r^{6}}{m^{3}})=2 +6\; log_8 \;r - 3 \;log_8\; m[/tex]

Learn more about the Logarithm here:

https://brainly.com/question/20785664

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