Question

What is the base 5 representation of the number 465^8

Answer 1

Answer:

$5^{30.57}$

Step-by-step explanation:

Let $x = 465^{8}$

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

log(x) = $log(465^{8})$

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

$\frac{logx}{log5}=\frac{log(465^{8} )}{log5}$

$log_{5}x=\frac{8log(465)}{log5}$

$log_{5}x$ = $\frac{8\times 2.66745}{0.69897}$

$log_{5}x$ = $\frac{21.339}{0.6989}$

$log_{5}x$ = 30.57

$x=5^{30.57}$

Therefore, base 5 representation of the given number will be $5^{30.57}$

Answer 2

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

$\ln \left(5^x\right)=\ln \left(465^8\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)$

$x\ln \left(5\right)=8\ln \left(465\right)$

$\mathrm{Divide\:both\:sides\:by\:}\ln \left(5\right)$

$\frac{x\ln \left(5\right)}{\ln \left(5\right)}=\frac{8\ln \left(465\right)}{\ln \left(5\right)}$

On simplifying, we get

$x=\frac{8\ln \left(465\right)}{\ln \left(5\right)}$

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