Expert: Paul Klarreich - 3/20/2010

Question
CONSIDER THE FOLLOWING SEQUENCES. WRITE DOWN THE NEXT TWO TERM OF EACH SEQUENCE

log (base 2) 8, log (base 4) 8. log (base 8)8, .....
log (base m) m^k, log (base m^2) m^k....

FIND THE EXPRESSION FOR THE N TERM OF EACH SEQUENCE. WRITE  THE EXPRESSION IN THE FORM OF P/q...

2. Log (base 4) 64, log (base 8) 64, log (base 32) 64
  Log(base 7)49, Log (base 49)49, log (base 343)49
  log (base 1/5) 125, log (base 1/125) 125. log(base1/625)125
describe how to obtain the third answer in each row from the first 2 answer.

Answer
Questioner: belle
Country: Malaysia
Category: Advanced Math
Private: No
Subject: mATH
Question: CONSIDER THE FOLLOWING SEQUENCES. WRITE DOWN THE NEXT TWO TERM OF

EACH SEQUENCE

log (base 2) 8, log (base 4) 8. log (base 8)8, .....
log (base m) m^k, log (base m^2) m^k....

FIND THE EXPRESSION FOR THE N TERM OF EACH SEQUENCE. WRITE  THE EXPRESSION

IN THE FORM OF P/q...

2. Log (base 4) 64, log (base 8) 64, log (base 32) 64
 Log(base 7)49, Log (base 49)49, log (base 343)49
 log (base 1/5) 125, log (base 1/125) 125. log(base1/625)125
describe how to obtain the third answer in each row from the first 2 answer
...........................................
Logarithm exercises always involve powers.  So write:

log (base 2) 8, log (base 4) 8. log (base 8)8, .....

as

log (base 2^1) 2^3, log (base 2^2) 2^3. log (base 2^3)2^3, .....

Do you get the pattern?
....................
And write:

log (base m^1) m^k, log (base m^2) m^k....

likewise, you should get the pattern.

.......................
Now then, for:

Log (base 4) 64, log (base 8) 64, log (base 32) 64

Did you skip a term?  Should it be:

Log (base 4) 64, log (base 8) 64, log (base 16) 64,...

which is:

Log (base 2^2) 2^6, log (base 2^3) 2^6, log (base 2^4) 2^6

So your n-th term should be

log (base 2^(n+1)) 2^6

Now what is that?  Write:

x = log (base 2^(n+1)) 2^6

and switch to exponential form:

(2^(n+1))^x = 2^6

Now solve:

2^[(n+1)x] = 2^6

(n+1)x = 6

x = 6/(n+1)

See if you can finish up now. Let me know if you run into trouble.