Question

What is the value of

Answer 1

\[\log_{81} 3\]

Answer 2

@UsukiDoll

Answer 3

@DecentNabeel

Answer 4

\[\log _{81}\left(3\right)=\frac{1}{4}\quad \left(\mathrm{Decimal:\quad }\:0.25\right)\]

Answer 5

@satellite73

Answer 6

Happy:) @EllenJaz17

Answer 7

how do you value that though? I have to show my work

Answer 8

@DecentNabeel

Answer 9

\[\mathrm{Rewrite\:}3\mathrm{\:\in\:power-base\:form:}\quad 3=81^{\frac{1}{4}}\]
\[=\log _{81}\left(81^{\frac{1}{4}}\right)\]
\[\mathrm{Apply\:the\:\log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)\]
\[\log _{81}\left(81^{\frac{1}{4}}\right)=\frac{1}{4}\log _{81}\left(81\right)\]
\[=\log _{81}\left(81\right)\frac{1}{4}\]
\[\mathrm{Apply\:the\:\log\:rule}:\quad \log _a\left(a\right)=1\]
\[\log _{81}\left(81\right)=1\]
=1/4

Answer 10

ok @EllenJaz17

Answer 11

\[\log_{81}(3)=x\iff 81^x=3\] so what you are trying to do is figure out how to write 3 as a power of 81
since \(\sqrt[4]{81}=3\) that tells you \[81^{\frac{1}{4}}=3\] and you have the power you are looking for

Answer 12

so it's not 1/4 its 3?

Answer 13

another easier example
\[\log_7(49)=2\] because \(7^2=49\)

Answer 14

no the answer is \(\frac{1}{4}\) let me write it again

Answer 15

ok sounds good

Answer 16

\[\huge \log_{81}(\color{red}x)=3\iff 81^{\color{red}x}=3\] you are trying to find \(\color{red}x\)

Answer 17

you are looking for the power that you would raise 81 to to give you 3

Answer 18

\[\log_2(8)=3\iff 2^3=8\] they say the same thing

Answer 19

you should be able to switch back and forth quickly

Answer 20

the only tricky part of this one was recognizing that 3 is the fourth root of 81

Answer 21

.sorry catching up was getting something to drink

Answer 22

beer i hope

Answer 23

haha no just water

Answer 24

too bad

Answer 25

I prefer straight tequila, gets the job done quick lol

Answer 26

no beer chaser? dang!

Answer 27

Haha nope!

Answer 28

oops
do you get that \(2^5=32\) says the same thing as \(\log_2(32)=5\) ?

Answer 29

how would i apply that to find this value in my question?

Answer 30

so if you were trying to solve
\[\log_2(32)=\color{red}x\] you would be trying to solve
\[2^{\color{red}x}=32\]

Answer 31

you have
\[\log_{81}(3)=\color{red}x\] you have to solve
\[81^{\color{red}x}=3\]

Answer 32

since \(\sqrt[4]{81}=x\) that is the same think as saying \(81^{\frac{1}{4}}=3\)

Answer 33

ooops i means "since \(\sqrt[4]{81}=3\)

Answer 34

now you have the power you are looking for , that power is \(\frac{1}{4}\) therefore since
\[81^{\frac{1}{4}}=3\] it is the same as saying
\[\log_{81}(3)=\frac{1}{4}\]

Answer 35

Ok i kinda get it lol

Answer 36

yeah it is confusing at first
a simpler example
is it clear that \(10^4=10,000\)?