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). If the base of the logarithm is negative, then the function is not continuous. 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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