Question

exponentials and logarithms are inverse functions.
consider the definition of exponential and logarithmic functions:
exponential function: $a = b^c$
logarithmic function: $\log_b(a) = c$
the variable $a$ is called the argument.
the variable $b$ is called the base
the variable $c$ is called the exponent
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

Explanation:

Step1: Identify variable roles in exponential function

For the exponential function \( a = b^c \):

• The variable \( a \) is the result of raising the base \( b \) to the exponent \( c \), so \( a \) is called the argument? Wait, no, in exponential form \( a = b^c \), \( b \) is the base, \( c \) is the exponent, and \( a \) is the power (or the result). But in the logarithmic function \( \log_b(a)=c \), \( a \) is the argument (the number we are taking the log of), \( b \) is the base, and \( c \) is the exponent (the result of the log). Wait, the problem says "The variable \( a \) is called the argument" – in the logarithmic function \( \log_b(a) = c \), \( a \) is the argument (the input to the log function, must be positive). For the exponential function \( a = b^c \), \( b \) is the base (must be positive and not 1), \( c \) is the exponent (can be any real number), and \( a \) is the output (so for exponential, \( a \) is the value, \( b \) is base, \( c \) is exponent). So the first dropdown: variable \( a \) is called the argument (matches logarithmic's argument, since log and exp are inverses). Then variable \( b \) is called the base (in both exp and log, \( b \) is the base, \( b>0, b

eq1 \)). Variable \( c \) is called the exponent (in exponential, \( c \) is the exponent; in logarithmic, \( c \) is the result, which is the exponent in exponential). So:

• \( a \): argument
• \( b \): base
• \( c \): exponent

Step2: Domain and range of \( a = b^c \)

For exponential function \( a = b^c \), where \( b>0, b
eq1 \), and \( c \) is any real number. The domain of \( a = b^c \) (treating \( c \) as the input variable) is all real numbers (since \( c \) can be any real number). The range of \( a = b^c \): since \( b>0 \), \( b^c \) is always positive (for any real \( c \)), and can take any positive real value (as \( c \) varies over reals, \( b^c \) covers \( (0, \infty) \)). So domain of \( a = b^c \) (input \( c \), output \( a \)): domain is \( (-\infty, \infty) \) (all real numbers), range is \( (0, \infty) \).

Step3: Domain and range of \( c = \log_b(a) \)

For logarithmic function \( c = \log_b(a) \), the argument \( a \) must be positive (so domain of \( a \) is \( (0, \infty) \)), and \( c \) can be any real number (since for any real \( c \), \( a = b^c \) is positive, so range of \( c = \log_b(a) \) is all real numbers \( (-\infty, \infty) \)). So:

• Domain of \( c = \log_b(a) \): \( a > 0 \) (so domain is \( (0, \infty) \))
• Range of \( c = \log_b(a) \): all real numbers \( (-\infty, \infty) \)

• Domain of \( a = b^c \): \( c \in \mathbb{R} \) (all real numbers)
• Range of \( a = b^c \): \( a > 0 \) (all positive real numbers)

So putting it all together:

1. Variable \( a \): argument (select "argument")
2. Variable \( b \): base (select "base")
3. Variable \( c \): exponent (select "exponent")

Domain of \( a = b^c \): all real numbers (since \( c \) can be any real number)
Range of \( a = b^c \): all positive real numbers ( \( a > 0 \) )

Domain of \( c = \log_b(a) \): \( a > 0 \) (all positive real numbers)
Range of \( c = \log_b(a) \): all real numbers ( \( c \in \mathbb{R} \) )

Answer:

• Variable \( a \): argument
• Variable \( b \): base
• Variable \( c \): exponent
• Domain of \( a = b^c \): All real numbers
• Range of \( a = b^c \): All positive real numbers
• Domain of \( c = \log_b(a) \): All positive real numbers
• Range of \( c = \log_b(a) \): All real numbers

(Assuming the dropdowns are filled with these: argument, base, exponent; then domain/range as above)