What is the domain of (ln(x-3))/(ln(e^x-e^3)) ?

The domain of (ln(x-3))/(ln(e^x-e^3)) is 3 ln(1+e^3)

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domain of (ln(x-3))/(ln(e^x-e^3))

domain

\frac{ln ( x - 3 )}{ln ( e ^{x} - e ^{3} )}

3 < x < ln (1 + e^{3}) or x > ln (1 + e^{3})

Interval Notation

(3, ln (1 + e^{3})) \cup (ln (1 + e^{3}), \infty)

Solution steps

Find positive values for logs:

x > 3

Find undefined (singularity) points:

x = ln (1 + e^{3})

Combine real regions and undefined points for final function domain

3 < x < ln (1 + e^{3}) or x > ln (1 + e^{3})

What is the domain of (ln(x-3))/(ln(e^x-e^3)) ?
The domain of (ln(x-3))/(ln(e^x-e^3)) is 3<x<ln(1+e^3)\lor x>ln(1+e^3)