Question

summary sheet #6: exponential and logarithmic functions
exponential function and it’s inverse:
graph $y = 3^x$ and it’s inverse on the grid provided and state the characteristics of the graphs.
(there is a graph of $y = 3^x$ with a table of x and y values: x=2,y=9; x=1,y=3; x=0,y=1; x=-1,y=1/3; x=-2,y=1/9. then a table for characteristics of exponential function and logarithmic function with rows: domain, range, y-int, equation of asymptote. then converting from exponents to logarithms table with exponential form and logarithmic form: $5^2 = 25$ with $\log_5 25 = 2$; $4^3 = 64$ with $\log_4 64 = 3$; $10^2 = 100$ with $\log 100 = 2$; $5^{\frac{1}{2}} = \sqrt{5}$ with $\log_5 \sqrt{5} = \frac{1}{2}$)

Explanation:

Step1: Analyze \( y = 3^x \) (Exponential Function)

• Domain: For any exponential function \( a^x \) (\( a>0, a

eq1 \)), \( x \) can be any real number. So domain of \( y = 3^x \) is \( (-\infty, \infty) \).

• Range: As \( x\to -\infty \), \( 3^x\to 0 \) (since \( 3^x=\frac{1}{3^{|x|}} \)), and as \( x\to \infty \), \( 3^x\to \infty \). So range is \( (0, \infty) \).
• y - int: Set \( x = 0 \), \( y=3^0 = 1 \). So y - intercept is \( (0, 1) \).
• Asymptote: As \( x\to -\infty \), \( y\to 0 \), so horizontal asymptote is \( y = 0 \).

Step2: Find Inverse of \( y = 3^x \) (Logarithmic Function)

To find the inverse, swap \( x \) and \( y \) and solve for \( y \). So \( x = 3^y \), which is \( y=\log_3 x \) (logarithmic function with base 3).

• Domain: For \( y = \log_3 x \), \( x>0 \) (since logarithm is defined for positive numbers). So domain is \( (0, \infty) \).
• Range: Logarithmic function \( \log_a x \) has range \( (-\infty, \infty) \) for \( a>0, a

eq1 \). So range is \( (-\infty, \infty) \).

• y - int: Set \( y = 0 \), \( 0=\log_3 x\implies x = 3^0 = 1 \). Wait, y - intercept is when \( x = 0 \), but \( \log_3 0 \) is undefined. So no y - intercept (or we can say it does not exist).
• Asymptote: As \( x\to 0^+ \), \( \log_3 x\to -\infty \), so vertical asymptote is \( x = 0 \).

Answer:

Characteristic	Exponential Function (\( y = 3^x \))	Logarithmic Function (\( y=\log_3 x \))
Range	\( (0, \infty) \)	\( (-\infty, \infty) \)
y - int	\( (0, 1) \)	Does not exist
Asymptote	Horizontal: \( y = 0 \)	Vertical: \( x = 0 \)

For the graph of \( y = 3^x \): It passes through \( (0, 1) \), \( (1, 3) \), \( (2, 9) \), \( (-1, \frac{1}{3}) \), \( (-2, \frac{1}{9}) \), increasing, approaches \( y = 0 \) as \( x\to -\infty \).

For the graph of \( y=\log_3 x \): It is the reflection of \( y = 3^x \) over the line \( y = x \), passes through \( (1, 0) \), \( (3, 1) \), \( (9, 2) \), \( (\frac{1}{3}, -1) \), \( (\frac{1}{9}, -2) \), increasing, approaches \( x = 0 \) as \( x\to 0^+ \).