Solution assignment 07 Logarithmic functions and graphs

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Assignment 7

Draw the graph of the function:

y=\log(x^2-7x+12)-\log(x-4)

Solution

The function exists if both logarithms exist. The first is defined for  x^2-7x+12>0, thus for x>4 or x<3. We know this because the quadratic function can be factorized:

x^2-7x+12=(x-4)(x-3)

The second logarithm is defined for x>4, thus the function as a whole is defined for x>4.

Now the function can be written as:

y=\log[(x-4)(x-3)]-\log(x-4)

=\log(x-4)+\log(x-3)-\log(x-4)

=\log(x-3)

We can sketch this graph more easily than the graph of the original function, but the graph is only valid for x>4:

log10(x-3)

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