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Logarithms and Exponential Functions

Logarithms and Exponential Functions

 Table of Contents

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