SOLUTION: what will be the value of log 0.3 to the base 9? the correct answer is -1/2. However i am not able to figure out the method

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: what will be the value of log 0.3 to the base 9? the correct answer is -1/2. However i am not able to figure out the method      Log On


   

Question 189805: what will be the value of log 0.3 to the base 9?
the correct answer is -1/2.
However i am not able to figure out the method
Found 2 solutions by jim_thompson5910, josmiceli:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Are you sure that the 0.3 isn't 0.3333 (where the 3's repeat)? If the 3's repeat, then 1%2F3=0.3333 (where the 3's repeat)



log%289%2C%280.3333%29%29 Start with the given expression


log%289%2C%281%2F3%29%29 Replace 0.3333 with 1%2F3


y=log%289%2C%281%2F3%29%29 Set the expression equal to "y"


9%5Ey=1%2F3 Use the property stated above to rewrite the equation


%283%5E2%29%5Ey=3%5E%28-1%29 Rewrite 9 as 3%5E2 and 1%2F3 as 3%5E%28-1%29


3%5E%282y%29=3%5E%28-1%29 Multiply the exponents


2y=-1 Since the bases are equal, the exponents are equal.


y=-1%2F2 Divide both sides by 2 to isolate "y"


Since y=log%289%2C%281%2F3%29%29, this means that log%289%2C%281%2F3%29%29=-1%2F2 and log%289%2C%280.3333%29%29=-1%2F2 (where the 3's repeat)

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
log%289%2C.3%29
This is saying that the base 9 raised to some
power gives me .3. I can rewrite that as:
9%5Ex+=+.3
9%5Ex+=+3%2F10
%283%5E2%29%5Ex+=+3%2F10
3%5E%282x%29+=+3%2F10
10%2A3%5E%282x%29+=+3
Now I can take the log to the base 10 of both sides
1+%2B+2x%2Alog%283%29+=+log%283%29
2x%2Alog%283%29+=+log%283%29+-+1
x+=+%28log%283%29+-+1%29+%2F+%282%2Alog%283%29%29
x+=+%28.477121+-+1%29+%2F+.954243
x+=+-.522879%2F.954243
x+=+-.547952
check answer:
9%5E-.547952+=+.29999976
This looks close enough
Actually,
9%5E%28-1%2F2%29+=+1%2F3 and not 3%2F10