Usually, but not necessarily.
A logarithm that is not an integer-value is irrational. For example log10100 = 2 which is a rational number. log1012 = 1.0791812460476... which is an irrational number.
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The number is called e, and it is approximately equal to 2.718.
There are an infinite number of irrational numbers. Here are some: e (the base for natural logarithms), pi, sqrt(2), sqrt(3), sqrt(5), square root of any number that is not a perfect square: perfect squares are 12 22 32 42 52 etc. which equals 1 4 9 16 25 ..... natural logarithm of any rational number (greater than zero) will be irrational. but not 1, since ln(1) = 0, which is not irrational. Note the logarithm of a negative number is a complex number, and the logarithm of zero is negative infinity.
Usually it is, yes. Of course, in some special cases the result of taking a logarithm is rational - such as taking the base-10 logarithm of 100.
Because it's an irrational number, and that's what "irrational" means. There are lots of other irrational numbers, like the base of the natural logarithm e or the square root of 2.In fact, there are more irrational numbers than rational numbers. A lot more.Infinitely more, even. There are an infinite number of rational numbers, but the infinite number of irrational numbers is a higher infinity than the infinity of rational numbers.
An irrational number.