The calculator reads a logarithm of the form y = log_b(x − h) + k. The result is the domain and the range, written in US interval notation and in a full sentence. The steps start from the rule that a real logarithm argument must be greater than 0.
Function y = logb(x − h) + k
Lesson: domain and range for a shifted log
A real logarithm log_b(u) is defined only when u is greater than 0. For y = log_b(x − h) + k the argument is x − h, so the inequality is x − h > 0, or x > h. In interval notation that set is (h, ∞). The endpoint h is excluded because the argument would be 0, and log_b(0) is not a real number.
The range of a real logarithm with a valid base is all real numbers. A vertical shift k moves the graph up or down but does not cut away any output values. Horizontal shift h moves the vertical asymptote, which changes the domain and leaves the range as (−∞, ∞).
This page does not plot the graph and does not solve log inequalities beyond that domain statement. To evaluate a single logarithm, use the Logarithm Calculator.
Practice problems
1. Common log, no shift
State the domain and range of y = log10(x).
2. Horizontal shift
State the domain and range of y = log2(x − 3) + 1.
3. Invalid base
Why does y = log1(x − 4) fail the calculator?
Show answers
Problem 1. Domain (0, ∞). Range (−∞, ∞).
Problem 2. Domain (3, ∞). Range (−∞, ∞).
Problem 3. Base 1 is not allowed for a real logarithm. The calculator reports that error instead of a domain.