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Log(0) & ln(0) Value – Simple Explanation


By log(x) we mean the logarithm of the number to the base 10. That is log_{10}(x) = y means that 10^y = x. On the other hand by ln(x) we mean the natural logarithm of the number to the base e. That is ln(x) = y means that e^y = x.

What is the value of ln(0)?

The value of ln(0) is equal to -\infty. Since -\infty is not actually a real number we sometimes also say that ln(0) is undefined or not defined. This is because of the following reason. If we assume that, ln(0) = y then it means that e^y = 0.

But the exponential function is always strictly positive and therefore there does not exist a real number y such that e^y = 0. Therefore we say that ln(0) is not well defined.

Also, note that if we put y = -\infty in the above equation we have that, e^{ -\infty} =\frac{1}{ e^\infty} = \frac{1}{ \infty}= 0. This gives a justification as to why we can say that ln(0) is equal to -\infty even though it is strictly speaking not a real number. A more precise way to mathematically say this would be that the limit of ln(x) approaches minus infinity as x tends closer and closer to 0.

What is the value of log10(0)?

By similar reasoning as above, we can also conclude that the logarithm of zero to the base 10 is also undefined or equal to minus infinity. Once again, this is because there is no real number satisfying the equation, 10^y = 0.

Once again, we see that as y tends closer and closer to minus infinity the value of 10y tends closer and closer to 0.

Is log(0) = 1?

The value of log(0) is not equal to 1. This is because if we assume that, log_{10}(0) = 1 then writing the above equation in exponential form we obtain that, 10^1 = 0. But, clearly 10 \neq 0 and so we conclude that the value of log(0) is not equal to 1.

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