For what value(s) of x, is it not possible to graph the function f(x)=log(x-1)? Explain, if possible, your reasoning with the help of a graph.

Answer 1

Log generally refers to the logarithm to the base 10.

For example:

10² = 100, so Log 100 = 2

All positive numbers greater than 0 can be written as an exponent of 10 and can represented on a graph.

For example;

2 equals 10^0.3010299... (therefore log 2 = 0.3010299...)

3 equals 10^{0.47712125...} (therefore log 3 = 0.47712125...)

and so on.

The graph has a vertical asymptote at 1, because 1-1 is 0 and there is no exponent of 10 that equals 0. There are also no exponents of 10 that equals a negative number. So the domain of f(x) = log (x-1) is (1, ∞); all values greater than 1

Answer 2

Euler's Identity is e^ix = cosx + isinx, which is true for all values of x

e^ipi =cospi + isinpi = -1 + i(0) = -1

e^ip=-1

take natural logs of both sides

ipi = ln(-1)

ln(-1) = ipi

while you can't graph this,(at least not on the Cartesian coordinate system with x and y axes measuring real numbers), as the natural log of negative numbers has no real solution, it does have an imaginary solution = non-real complex value = about 3.14159 times the square root of negative one.

same is true for the common log to the base 10 or to any base. logs of non-positive numbers cannot be graphed So, log(x-1) has no defined value (for real numbers) if x<1, But for x=0, ln(0-1) = ipi. It does have a non-real complex defined value = i(pi)

Answer 3

Hello, Anne,

At x=1 the function becomes undefined. The expression x-1 becomes just zero, 0. log (0) is asking what exponent of 10 will result in 0. 10⁰ is 1. There are no exponents of 10 that will produce zero.

Bob