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Limits of Logarithmic Functions

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

Let's hold up the mirror by taking the base-4 logarithm to get the inverse function: f (x) = log4 x.

If we tried to make x negative or zero in this log function, 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. In other words, its domain is x > 0.

Here's what the graph of f (x) = log4 x looks like:

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

Sample Problem

When will f (x) = log5 x be greater than g (x) = 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, we get 50.43 or 200.23.
For x = 5, we get 51 or 200.54.

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  , we get 5-0.43 or 20-0.23.

For  , we get 5-0.86 or 20-0.46.

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