Question

(the v means downards, these 3's are subscripted)
solve for x: logv3(x-8) + logv3(x) = 2
Please be specific! Thank you! :)

Answer 1

underscore means subscript

Answer 2

\[\log_3(x-8) + \log_3(x) = 2\]
\[=\log_3((x-8)x)=2\]

Answer 3

now use the definition of logarithm
\[\log_bx=y\qquad\rightarrow\qquad b^y=x\]

Answer 4

\[3^2=(x-8)x\]

Answer 5

Since this is a logarithm is is it 2x-8x = 9 or x^2-8x = 9? Is the final answer = 2/3? If not, how do I solve x^2-8x = 9?

Answer 6

\[x^2-8x = 9\]\[x^2-8x -9=0\]\[=(x+...)(x-...)\]

Answer 7

looking for two numbers that have a difference of 8 and a product of -9

Answer 8

now there product is negative so one of the numbers must be negative

Answer 9

x = 9 x = -1right?

Answer 10

x = -1 is extraneous?

Answer 11

\[x^2−8x−9=0\]
\[(x+1)(x-9)=0\]
\[x=-1,9\]

Answer 12

x=9 in first part and x=-1 in second is illegal

Answer 13

in real log we can not chose negative variable

Answer 14

\[\log_3(x-8) + \log_3(x) = 2\]
if we substitute x=-1 in to original equation
\[\log_3(-1-8) + \log_3(-1) = 2\]
\[\log_3(-9) + \log_3(-1) = 2\]
but you cannot take the log of a negative number

if we substitute x=9 in to original equation
\[\log_3(9-8) + \log_3(9) = 2\]\[\log_3(1) + \log_3(9) = 2\]\[\log_3(9)=2\]\[3^2=9\]which is true

Answer 15

ok i guess it was 8-x in log
so only x=9 is answer

Answer 16

Thank you so much to both of you guys!! This helped a lot :)