Question

Express y as a function of x. What is the domain? log4y = (x+log4)/4

Answer 1

Why is the answer y = 10^x and domain is real numbers?

Answer 2

writing the equation in exponential form you have:

\(\large 10^{\frac{x+log4}{4}}=4y \)

Answer 3

how'd you know it would be 10...

Answer 4

logs written without a base is log base 10..

Answer 5

oh yeah

Answer 6

and i'm assuming the left side of your equation is \(\large log(4y) \) and not \(\large log_4y \)

Answer 7

yes you're correct

Answer 8

ohh shoot, it's only:

Log4y = x + log4

Answer 9

no... \(\large log(4y)=log4+logy \)
if you're simplifying the left side of your equation first..

Answer 10

I meant the original question is log4y = x + log4... what happened to the x?

Answer 11

that last post I put is just simplifying the left side of your original equation...
is this you original equation:
\(\large log(4y)=\frac{x+log4}{4} \) or \(\large log(4y)=x+log4 \)

Answer 12

ohh.. the second one

Answer 13

i'm guessing it is the second one and not the first because you said the answer is y=10^x....
this makes it much simpler....

the original equation is: \(\large log(4y)=x+log4 \)

so using the properties of logs on the left side of this equation you have
log(4y) = log4 + logy
so the equation is now: \(\large log4+logy=x+log4 \)
cancelling log4 from both sides you have: \(\large \color {red}{log4}+logy=x+\color {red}{log4} \)
so

\(\large logy=x \)

now this can be written in exponential form: \(\large y=10^x \)

Answer 14

I don't understand this part: log(4y) = log4 + logy

Answer 15

there is a property (a very useful one):
\[\huge log_b(M\cdot N)=log_bM + log_bN \]

Answer 16

ohh I know that one but what happened to the x-- you replaced it with log y..

Answer 17

ohh, never mind. I get it now, lol thank you! :D

Answer 18

ok.. let's start from the beginning with the original, correct equation:
\[\large \color {red}{log(4y)}= x+log4\]

using the property on the left side of the equation we get:

\[\large \color {red}{log4+logy}= x+log4\]

you ok so far?

Answer 19

yes I'm okay with it so far

Answer 20

do I transpose log 4 to the right side?

Answer 21

yes... subtract log4 from both sides of the equation, you have \(\large logy=x \) left.

now just write this equation out in exponential form to get \(\large 10^x = y\)

Answer 22

ohh, by the way.. when do I divide logs? I know that if I subtract logs then I divide them correct?

Answer 23

the property you're referring to is this: \[\huge log_b(\frac{M}{N})=log_bM-log_bN \]

Answer 24

the log of a quotient(division) is the difference of logs....

Answer 25

yes that one. So I only divide when they're both different logs?

Answer 26

"difference" meaning subtraction....

Answer 27

yeah I know, so in this case the log4-log4 -- I can divide?

Answer 28

log4 is just some number.... so a number minus itself is zero...

Answer 29

ohh.. so when exactly do I use the rule about dividing differences..?

Answer 30

and yeah you're right too: log4 - log4 = log(4/4) = log(1) = 0

Answer 31

ohhh, okay. thank you!!

Answer 32

just a longer way to the simplified answer...

Answer 33

yw...:)