what is the domain and range of log(tanx)?
thanks! :)
Found this online:
"The domain of a logarithmic function is real numbers greater than zero, and the range is real numbers. The graph of y = logax is symmetrical to the graph of y = ax with respect to the line y = x. This relationship is true for any function and its inverse."
Is this helpful?
As noted above, you need tanx > 0
So, for what values of x is tanx > 0 ?
And of course, the range of logx is all real numbers
Range is all real numbers
To find the domain and range of the function log(tan(x)), we need to consider the properties and restrictions of the logarithmic and tangent functions.
1. Domain:
The domain of the logarithmic function is defined as all positive real numbers. However, for the tangent function, there are some restrictions.
The tangent function repeats its values every π radians, resulting in vertical asymptotes at odd multiples of π/2 (i.e., x = π/2, 3π/2, 5π/2, etc.). Therefore, we need to exclude these values from the domain since tan(x) would be undefined at those points where the tangent function approaches infinity or negative infinity.
So, the domain of log(tan(x)) is all x-values where x is not equal to π/2, 3π/2, 5π/2, etc.
2. Range:
The range of the logarithmic function is defined as all real numbers, and the range of the tangent function is all real numbers except for values where the function approaches infinity or negative infinity.
As the logarithmic function takes the logarithm of the tangent function, we can conclude that the range of log(tan(x)) is all real numbers.
In summary:
Domain: All real numbers except (odd multiples of π/2)
Range: All real numbers
Please note that this explanation assumes we are working with the natural logarithm (base e) unless specified otherwise.