In other words, there are gaps between the integer powers where the function is only defined in the nonreal numbers. However, what about values between the integers? For instance, what if I asked you what power I needed to raise -2 to in order to get 1/2? The answer is a complex number, and it can only be found with some knowledge of trigonometry and the de'Moivre's theorem. For instance, sure the logarithm is defined for even and odd powers of negative numbers (though even powers are positive and the odd powers a negative and this is a wild jumping behavior that will continue for all integers).
Calculate the exponent and mantissa for each number. Create a vector X that contains several test values. These operations all follow standard IEEE® arithmetic. If the base of the logarithm is negative, then the function is not continuous. Dissect several numbers into the exponent and mantissa. We can never have 1ˣ = 2 or 1ˣ = 938 or 1ˣ = any number besides 1. Logarithms are undefined for base 1 because there exist no real power that we could raise one to that would give us a number other than 1. So, it follows that b ≠ 1 b\neq1 b = 1 b, does not equal, 1. But this can never be true since 1 1 1 1 to any power is always 1 1 1 1. The equivalent exponential form would be 1 x = 3 1^x=3 1 x = 3 1, start superscript, x, end superscript, equals, 3. Now consider the equation log 1 ( 3 ) = x \log_1(3)=x lo g 1 ( 3 ) = x log, start base, 1, end base, left parenthesis, 3, right parenthesis, equals, x. Suppose, for a moment, that b b b b could be 1 1 1 1. ī ≠ 1 b\neq1 b = 1 b, does not equal, 1
Because a positive number raised to any power is positive, meaning b c > 0 b^c>0 b c > 0 b, start superscript, c, end superscript, is greater than, 0, it follows that a > 0 a>0 a > 0 a, is greater than, 0. Log b ( a ) = c \log_b(a)=c lo g b ( a ) = c log, start base, b, end base, left parenthesis, a, right parenthesis, equals, c means that b c = a b^c=a b c = a b, start superscript, c, end superscript, equals, a. In an exponential function, the base b b b b is always defined to be positive. Log 2 ( 8 ) = 3 \log_\blueD2(\goldD 5 2 = 2 5 start color #11accd, 5, end color #11accd, start superscript, start color #1fab54, 2, end color #1fab54, end superscript, equals, start color #e07d10, 25, end color #e07d10
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