Question

What are the domain and range of f (x) = log (x minus 1) 2?.

Answer 1

You can use the definition of logarithm and the fact that a positive number raised to any power will always stay bigger than 0.

The domain of the given function is  {x | x > 1 and a real number }

The range of the given function is $\mathbb R$ (set of real numbers)

What is the definition of logarithm?

If a is raised to power b is resulted as c, then we can rewrite it that b equals to the logarithm of c with base a.

Or, symbolically:

$a^b = c \implies b = log_a(c)$

Since c was the result of a raised to power b, thus, if a was a positive number, then a raised to any power won't go less or equal to zero, thus making c > 0

How to use this definition to find the domain and range of given function?

Since log(x-1) is with base 10 (when base of log isn't specified, it is assumed to be with base 10) (when log is written ln, it is log with base e =2.71828.... ) thus, we have a = 10 > 0 thus the input x-1 > 0 too.

Or we have:

x > 1 as the restriction.

Thus domain of the given function is {x | x > 1 and a real number }

Now from domain, we have:

$x > 1\\ x-1 > 0\\ log(x-1) > -\infty\\ log(x-1) + 2 > -\infty\\ f(x) > -\infty$ (log(x-1) > -infinity since log(0) on right side have arbitrary negatively large value which is denoted by -infinity)

Thus, range of given function  is whole real number set $\mathbb R$ (since all finite real numbers are bigger than negative infinity)

Thus, the domain of the given function is  {x | x > 1 and a real number }

The range of the given function is $\mathbb R$ (set of real numbers

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