Question

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

Answer 1

`5^(30.57)`

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.

`(logx)/(log5)=(log(465^(8) ))/(log5)`

`log_(5)x=(8log(465))/(log5)`

`log_(5)x` =
`(8* 2.66745)/(0.69897)`

`log_(5)x` =
`(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)`

`(x\ln \left(5\right))/(\ln \left(5\right))=(8\ln \left(465\right))/(\ln \left(5\right))`

On simplifying, we get

`x=(8\ln \left(465\right))/(\ln \left(5\right))`

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