In relation to exponential and logarithmic functions, create a function that has the following key features (show all work in forming function to model features): * Increasing for x > 0 *...

In relation to exponential and logarithmic functions, create a function that has the following key features (show all work in forming function to model features):

* Increasing for x > 0

* x-intercept of 1

* no y-intercept

* a vertical asymptote of x=0

* Domain: {XER | x>0}

* Range: {YER}

* (5,1) is a point of the graph

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First we must determine whether this is logarithmic or exponential.

All exponential functions have a y-intercept so this must be logarithmic.

The general form for a logarithmic function is `y=alog_b (x-h)+k`

 

The domain for this function is `x>h, x in RR` .

The function is increasing on its domain.

There is no y-intercept if h>0.

There is a vertical asymptote at x=0 if h=0.

The range is all real numbers.

So with h=0 we have `y=alog_bx + k` .

If the x-intercept is 1 then `log_b 1=0` as long as there is no horizontal or vertical translation.

The point (5,1) is on the graph so when x=5 y=1:

Let the base be 5. (`log_5 5=1` so the point (5,1) is on the graph if there is no horizontal or vertical translation.)

Then one function we can use is `y=log_5 x`

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One function that fits the requirements is `y=log_5 x`

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The graph:

` `

 

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