# If we are given log 81 = 4, can we find the value of log 27?

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### 2 Answers

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.