Question

how do you do this

Answer 1

\[\log_b(x)=y\iff b^y=x\] so if \[\log_4(x)=2.5\] what is \(x\)?

Answer 2

with guess and check i found it to be 32

Answer 3

my problem is how do i do it without guess and check

Answer 4

you sort of have to know it \[4^{2.5}=4^{\frac{5}{2}}=\sqrt{4^5}\]

Answer 5

since \(\sqrt4=2\) you have \[\sqrt{4^5}=2^5\]

Answer 6

that is just \(x\) of course, you still have to find \(y\) to finish the question

Answer 7

yah

Answer 8

but i don;t understand how to find y cause that one is hard to do guess and check

Answer 9

so you have to kind of guess this one too \[y^{-\frac{3}{2}}=125\]

Answer 10

well not really \[y^{-\frac{3}{2}}=125\iff y=125^{-\frac{2}{3}}\]

Answer 11

and \[125^{-\frac{3}{2}}=\frac{1}{\sqrt[3]{125}^2}\]

Answer 12

oops sorry, typo there, i meant \[\huge 125^{-\frac{2}{3}}=\frac{1}{\sqrt[3]{125}^2}\]

Answer 13

so to find y you have to do 125^-2/3 ?

Answer 14

yes exactly
which is not hard

Answer 15

.04

Answer 16

so it is all 800

Answer 17

i guess, it is as a fraction \[\frac{1}{25}\]

Answer 18

whatever \(32\times 25\) is yes

Answer 19

wait can i ask u how to find the first part of the log to be 32 without guessing

Answer 20

yes, we started with \[4^{2.5}\] right ?

Answer 21

oh yes

Answer 22

as a fraction \(2.5=\frac{5}{2}\) yes ?

Answer 23

yeah i see thanks :)

Answer 24

yw