Aptitude - Logarithm - Discussion

Logarithm base change rule
The base b logarithm of x is base c logarithm of x divided by the base c logarithm of b.

logb(x) = logc(x) / logc(b)

For example, in order to calculate log2(8) in calculator, we need to change the base to 10:

log2(8) = log10(8) / log10(2)

So by this we get
log5 512 = log 512 / log 5 with base 10

Now since 5 can be written as 10/2
so log 5 = log(10/2)

Now using property log(x / y) = log(x) - log(y)
so, log(10/2) can be written as log(10) - log(2)

So finally we got
log5 512 = log(512) / log(10) - log(2)
since 512 = 2^9

log 512 = log(2^9) = 9 log 2

= 9 log 2 / log 10 - log 2
= (9 x 0.3010) / 1 - 0.3010
= 2.709 / 0.699 = 3.876

Hope it helps !

How can log 10 be 1? It can only be 1 if the base is 10. If I remember the formula correctly then the base can be anything and is generally a random constant term (a, b etc. ). If we follow the same logic then the answer is most definitely wrong as per my calculations.

CAN ANYONE EXPLAIN THE CORRECT PROCESS?

@Kasthuri

log 75= log(5*5*3) = log(5*5)+log(3), because log m*n=log m+log n
=2 log5 +log3
=2 log(10/2)+log3
=2(log10-log2)+log3
=2(1-.3010)+..4771
=2*.699+.4771
=1.398+.4771
=1.8751

log5512 = log512/log5 = 9log2/log5.
= 9log2/log3+log2.
= 9*0.3010/0.4771+0.3010.

I think this is this the correct procedure as log 2 and log3 values are given. Why we need to go for log 10/2?

log(75) = log(70+5).

= log(7*10)+(2+3).

= [log(7)+log(10)]*log(2)*log(3).

= [0.8541+1]*0.3010*0.4771.

= 1.8451*0.3010*0.4771.

= 0.1557.

@Dinesh Puri.

log 4=log 2^2.
=2log2,
=2log(10/5),
= 2*{log10-log5},
= 2*{1-.690},
= 0.62.

@jeromy.

Click the impotant formulas link there you can see 7th formula using that formula we divide log512 by log5.

That is base change rule it seems log (b) to the base a, = log (b) to base c/log (a) to base c i.e log 512/log5.

No @Amrita.

Log 5 can't be written as log 2+log3.
But it can be written as log 5+ log 1.
Hope it helps.

@Nirav.

log512/log5 = 9log2/log(2+3) = 9log2/(log2*log3).

= 9*.3010/(.3010*.4771).

= 9/.4771 >18.