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Limits of Logarithmic Functions

Just like exponential functions, log functions have their own limits. Remember what exponential functions can't do: they can't output a negative number. The function we took a gander at when thinking about exponential functions was y = 4x.

Let's hold up the mirror by taking the base-4 logarithm to get logx = y. If we tried to make x negative or zero in either of these two equivalent functions, there is no y value in the known universe that would let us do it—so the log function is "undefined" at x values of zero or less.

Because the output of an exponential can never be zero or negative, the inverse (log) function can never have a zero of negative input.

Sample Problem

When will log5 x be greater than log20 x? Ignore negative x values.

The output of these logs is the exponent needed above 5 or 20 to equal x.

When x is greater than 1, log5 x will be larger because a larger exponent is needed for 5 than for 20 to equal whatever x is in this domain. The 5 is a little shrimp that needs a boost, basically. For example:

For x = 2 ; 500.431or 2000.231
For x = 5; 51 or 2000.537

However, when x is less than 1, log20 x will be greater than log5 x because a negative exponent closer to zero is needed to pull that 20 down to size versus the 5. That 20 has a lot of muscle, and needs a big push by the exponent to get down to size.

For x =  ; 5-0.43 or 20-0.23

For x =  ; 5-0.86 or 20-0.46

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