Question

the domain of a = b^c is
the range of a = b^c is
the domain of c = log_b(a) is
the range of c = log_b(a) is
question help: ▶ video
submit all parts
basic interval
enter an interval using interval notation

Explanation:

Step1: Recall properties of exponential and logarithmic functions

For the exponential function \(a = b^c\) (assuming \(b>0, b
eq1\)):

• The domain (set of all possible \(c\)-values) is all real numbers because we can raise \(b\) to any real power. So the domain of \(a = b^c\) is \((-\infty, \infty)\).
• The range (set of all possible \(a\)-values) is \( (0, \infty)\) because an exponential function with \(b>0, b

eq1\) always gives positive outputs (it never equals zero or is negative).

For the logarithmic function \(c=\log_b(a)\) (assuming \(b > 0, b
eq1\)):

• The domain (set of all possible \(a\)-values) is \( (0, \infty)\) because we can only take the logarithm of a positive number.
• The range (set of all possible \(c\)-values) is \((-\infty, \infty)\) because we can get any real number as the output of a logarithm (for example, \(\log_b(b^k)=k\) for any real \(k\)).

Step2: Fill in the intervals

• Domain of \(a = b^c\): \((-\infty, \infty)\)
• Range of \(a = b^c\): \((0, \infty)\)
• Domain of \(c=\log_b(a)\): \((0, \infty)\)
• Range of \(c=\log_b(a)\): \((-\infty, \infty)\)

Answer:

s (for each part):

• Domain of \(a = b^c\): \(\boldsymbol{(-\infty, \infty)}\)
• Range of \(a = b^c\): \(\boldsymbol{(0, \infty)}\)
• Domain of \(c=\log_b(a)\): \(\boldsymbol{(0, \infty)}\)
• Range of \(c=\log_b(a)\): \(\boldsymbol{(-\infty, \infty)}\)