If we are given log 81 = 4, can we find the value of log 27?
We are given that log 81 = 4. We are not given the base of the logarithm here.
For a logarithm to the base b, if log(b) a =c it follows that a = b^c.
We use this relation here, let the base of the logarithm be n.
Therefore log(n) 81 = 4
=> 81 = n^4
=> 3^4 = n^4
Therefore we have determined that n is equal to 3. Here n could also have been -3 but we have to remember that if the base of a log is negative, the logarithm is defined only in certain cases and not for all numbers. So we take only +3.
Therefore log (3) 27 = log (3) 3^3 = 3* log (3) 3 = 3.
The value of log 27 is 3.
If log81 = 4, then to find log 27.
We know by the definition of logarithms that:-
If a^x = y, then log a(y) = x.
Alternatively, if loga (y) = x, then a^x = y.
Also this means log(a) a^x = x .
Since log81 = 4.
So log(3) 81 = 4, log(3) 3^4 = 4
Therefore log (3)27 = log (3) 3^3 .
Therefore log (27) = 3.