Respuesta :

We need to convert given value 465^8 as the base 5 number.

Let us assume x is the exponent of 5 when we take base 5.

We can setup an equation now.

5^x=465^8

Taking ln on both sides, we get

[tex]\ln \left(5^x\right)=\ln \left(465^8\right)[/tex]

[tex]\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)[/tex]

[tex]x\ln \left(5\right)=8\ln \left(465\right)[/tex]

[tex]\mathrm{Divide\:both\:sides\:by\:}\ln \left(5\right)[/tex]

[tex]\frac{x\ln \left(5\right)}{\ln \left(5\right)}=\frac{8\ln \left(465\right)}{\ln \left(5\right)}[/tex]

On simplifying, we get

[tex]x=\frac{8\ln \left(465\right)}{\ln \left(5\right)}[/tex]

Therefore, [tex]465^8 \ can \ be \ written \ as \ 5^{\frac{8\ln \left(465\right)}{\ln \left(5\right)}}.[/tex].

Answer:

[tex]5^{30.57}[/tex]

Step-by-step explanation:

Let [tex]x = 465^{8}[/tex]

Now we take log on both the sides of the equation.

log(x) = [tex]log(465^{8})[/tex]

Now to get 5 as the base we will divide this equation by log5 on both the sides.

[tex]\frac{logx}{log5}=\frac{log(465^{8} )}{log5}[/tex]

[tex]log_{5}x=\frac{8log(465)}{log5}[/tex]

[tex]log_{5}x[/tex] = [tex]\frac{8\times 2.66745}{0.69897}[/tex]

[tex]log_{5}x[/tex] = [tex]\frac{21.339}{0.6989}[/tex]

[tex]log_{5}x[/tex] = 30.57

[tex]x=5^{30.57}[/tex]

Therefore, base 5 representation of the given number will be [tex]5^{30.57}[/tex]