Question

topic 5: converting exponential and logarithmic equations

1. label each as exponent, argument, or base.

a. \\(\log_{4} 64 = 3\\)
4 is the _______
64 is the _______
3 is the _______

b. \\(5^2 = 25\\)
5 is the _______
2 is the _______
25 is the _______

1. convert each from one form to the other.

exponential form	\\(\leftrightarrow\\)	logarithmic form
\\(3^4 = y\\)	\\(\leftrightarrow\\)
	\\(\leftrightarrow\\)	\\(\log 1000 = 3\\)
\\(e^x = 54.6\\)	\\(\leftrightarrow\\)

topic 6: exponential and logarithmic graphs

1. label each as an exponential or logarithmic graph. state how you know.

Explanation:

Identify components of logarithmic and exponential equations

We define the parts of logarithmic and exponential equations to label Question 14.
For a logarithm \(\log_{b}(a) = c\):

• \(b\) is the BASE
• \(a\) is the ARGUMENT
• \(c\) is the EXPONENT (or logarithm value)

For an exponential equation \(b^c = a\):

• \(b\) is the BASE
• \(c\) is the EXPONENT
• \(a\) is the ARGUMENT (the result of raising the base to the exponent)

Applying this to:
a. \(\log_{4}64 = 3\)

• \(4\) is the BASE
• \(64\) is the ARGUMENT
• \(3\) is the EXPONENT

b. \(5^2 = 25\)

• \(5\) is the BASE
• \(2\) is the EXPONENT
• \(25\) is the ARGUMENT

Convert between exponential and logarithmic forms

We use the equivalence relation \(b^c = a \iff \log_{b}a = c\) to complete the table in Question 15:

• Row 1: Given Logarithmic Form \(c = \log_{b}a\). The equivalent Exponential Form is \(b^c = a\).
• Row 2: Given Exponential Form \(3^4 = y\). The equivalent Logarithmic Form is \(\log_{3}y = 4\).
• Row 3: Given Logarithmic Form \(\log 1000 = 3\) (common logarithm with base \(10\)). The equivalent Exponential Form is \(10^3 = 1000\).
• Row 4: Given Exponential Form \(e^x = 54.6\). The equivalent Logarithmic Form is \(\ln(54.6) = x\) or \(\log_{e}(54.6) = x\).

Analyze and classify the graphs

We identify the characteristic shapes and asymptotes of exponential and logarithmic functions to label the graphs in Question 16:

• Exponential graphs of the form \(y = b^x\) have a horizontal asymptote (usually \(y=0\)) and a domain of all real numbers \((-\infty, \infty)\).
• Logarithmic graphs of the form \(y = \log_b(x)\) have a vertical asymptote (usually \(x=0\)) and a domain restricted to positive real numbers \((0, \infty)\).

Let's classify each of the four graphs from left to right:

1. First Graph: Has a horizontal asymptote on the left and grows rapidly on the right. This is an exponential graph.
2. Second Graph: Has a vertical asymptote on the left and grows slowly to the right. This is a logarithmic graph.
3. Third Graph: Has a horizontal asymptote on the right and grows rapidly to the left. This is a decaying exponential graph.
4. Fourth Graph: Has a vertical asymptote on the left and decreases to the right. This is a decaying logarithmic graph.

Answer:

Question 14

a. \(\log_{4}64 = 3\)

• 4 is the BASE
• 64 is the ARGUMENT
• 3 is the EXPONENT

b. \(5^2 = 25\)

• 5 is the BASE
• 2 is the EXPONENT
• 25 is the ARGUMENT

Question 15

Exponential Form	\(\iff\)	Logarithmic Form
\(3^4 = y\)	\(\iff\)	\(\log_{3}y = 4\)
\(10^3 = 1000\)	\(\iff\)	\(\log 1000 = 3\)
\(e^x = 54.6\)	\(\iff\)	\(\ln(54.6) = x\)

Question 16

• First Graph: Exponential graph. We know this because it has a horizontal asymptote (\(y = 0\)) and a domain of all real numbers.
• Second Graph: Logarithmic graph. We know this because it has a vertical asymptote (\(x = 0\)) and a restricted domain of positive real numbers.
• Third Graph: Exponential graph. We know this because it has a horizontal asymptote (\(y = 0\)) and a domain of all real numbers.
• Fourth Graph: Logarithmic graph. We know this because it has a vertical asymptote (\(x = 0\)) and a restricted domain of positive real numbers.