🎯⭐ INTERACTIVE LESSON

Logarithmic Functions and Equations

Learn step-by-step with interactive practice!

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Logarithmic Functions and Equations - Complete Interactive Lesson

Part 1: Introduction to Logarithms

📖 Logarithmic Functions — Definition & Inverse Relation

Part 1 of 7

A logarithm answers: "What exponent do I need?"

$\boxed{\log_b x = y \iff b^y = x}$

" $\log$ base $b$ of $x$ equals $y$ " means " $b$ raised to $y$ gives ."

The Three Standard Bases

Notation	Name	Base	Calculator Key
$\log x$	Common log	$10$	LOG
$\ln x$	Natural log	$e \approx 2.718$

Quick Conversion Examples

Exponential Form	Logarithmic Form
$2^5 = 32$	$\log_2 32 = 5$

🔄 Logs and Exponentials Are Inverses

If $f(x) = b^x$ , then $f^{-1}(x) = \log_b x$ .

📝 Worked Example: Converting & Evaluating

Evaluate $\log_4 64$ without a calculator.

Step 1: Rewrite as an equation: $\log_4 64 = x$ means

Concept Check 🎯

Evaluate These Logs 🧮

1) $\log_5 125 =$ ? (e.g., $\log_2 64$ : since $^{}$ , the answer is )

Log Fundamentals 🔽

Exit Quiz ✅

Part 2: Properties of Logarithms

📐 Logarithmic Functions — Core Log Properties

Part 2 of 7

Log properties turn multiplication, division, and exponentiation into addition, subtraction, and scalar multiplication.

The Three Fundamental Properties

Property	Rule	Direction
Product Rule	$\log_b(xy) = \log_b x + \log_b y$

Part 3: Solving Logarithmic Equations

📈 Logarithmic Functions — Transformations & Graphs

Part 3 of 7

The general transformed logarithmic function:

$\boxed{g(x) = a \cdot \log_b(x - h) + k}$

Part 4: Change of Base

🔍 Logarithmic Functions — Solving Log Equations

Part 4 of 7

The Two Core Strategies

Strategy	When to Use	Key Move
Rewrite as exponential	Single log on one side	$\log_b(\text{stuff}) = c \implies \text{stuff} = b^c$

Part 5: Logarithmic Models

🔀 Logarithmic Functions — Change of Base & Calculator Fluency

Part 5 of 7

Calculators only have LOG ( $\log_{10}$ ) and LN ( $\ln$ ) keys. To evaluate any other base, use the Change of Base Formula:

$\log_{b} x = \frac{\log x}{\log b} = \frac{}{}$

Part 6: Problem-Solving Workshop

🌍 Logarithmic Functions — Modeling with Logs

Part 6 of 7

Logarithmic scales appear throughout science. They compress enormous ranges into manageable numbers.

Real-World Log Scales

Scale	Formula	What It Measures
pH	$\text{pH} = -\log[H^+]$	Acidity (hydrogen ion concentration)
Decibels

Part 7: Review & Applications

🏆 Logarithmic Functions — Full Synthesis

Part 7 of 7 — Putting It All Together

Your Complete Log Toolkit

Concept (Part)	Key Idea
Definition & Inverse (1)	$\log_b x = y \iff b^y = x$ ; domain

Inverse Properties — they undo each other:

$\boxed{\log_b(b^x) = x \quad \text{and} \quad b^{\log_b x} = x}$

Expression	Simplifies To	Why
$\log_2(2^7)$	$7$	Log undoes the exponential
$10^{\log 50}$	$50$	Exponential undoes the log
$\ln(e^{-3})$	$-3$	$\ln$ undoes $e^x$
$e^{\ln 12}$	$12$	$e^x$ undoes $\ln$

Graphical Connection

The graph of $y = \log_b x$ is the reflection of $y = b^x$ over the line $y = x$ .

Feature	$y = b^x$ (with $b > 1$ )	$y = \log_b x$
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$($

Step 2: Express both sides as powers of $4$ :

$4^1 = 4$
$4^2 = 16$
$4^3 = 64$ ✔

$\boxed{\log_4 64 = 3}$

Pattern for Evaluating Logs Mentally

Step	Action
1	Set $\log_b x = ?$ → rewrite as $b^? = x$
2	Find common base or multiply repeatedly
3	Match exponents

Key Values to Memorize

2) $\log_{10} 0.001 =$ ? (e.g., $\log_{10} 0.01$ : since $10^{-2} = 0.01$ , the answer is $-2$ )

3) $\ln(e^5) =$ ? (e.g., $\ln(e^{-2})$ : by inverse property, the answer is $-2$ )

⚠️ Critical restriction: These rules only apply to products, quotients, and powers inside the log. There is no rule for $\log(x + y)$ or $\log(x - y)$ .

🔓 Expanding Logarithmic Expressions

"Expanding" means using the rules left-to-right to break a single log into simpler pieces.

Worked Example 1

Expand $\log_3\left(\frac{x^2 y}{z^4}\right)$

Step	Action	Result
1	Quotient rule	$\log_3(x^2 y) - \log_3(z^4)$

$\boxed{\log_3\left(\frac{x^2 y}{z^4}\right) = 2\log_3 x + \log_3 y - 4\log_3 z}$

Order of Operations for Expanding

Quotient rule first (handle the fraction)
Product rule next (break up any remaining products)
Power rule last (pull exponents out front)

🔒 Condensing Logarithmic Expressions

"Condensing" means running the rules right-to-left to combine multiple logs into one.

Worked Example 2

Condense $3\ln a - \frac{1}{2}\ln b + \ln c$

Step	Action	Result
1	Power rule (reverse)	$\ln a^3 - \ln b^{1/2} + \ln c$

$\boxed{3\ln a - \frac{1}{2}\ln b + \ln c = \ln\left(\frac{a^3 c}{\sqrt{b}}\right)}$

Why Condensing Matters

Solving equations requires one log on each side
Condensing to a single log lets you drop the log and solve the argument

🚫 Common Errors to Avoid

❌ Wrong	✅ Correct	Why
$\log(x + y) = \log x + \log y$	No simplification exists	Log of a sum ≠ sum of logs
$\log(x - y) = \log x - \log y$	No simplification exists	Log of a difference ≠ difference of logs
$(\log x)^2 = 2\log x$	$(\log x)^2$ stays as is
$\frac{\log x}{\log y} = \log\frac{x}{y}$

Quick Memory Aid

The rules work for operations inside the log argument:

Inside multiplication → product rule

Inside division → quotient rule

Inside exponent → power rule

If it's addition or subtraction inside, stop — no rule applies.

Properties Quiz 🎯

Expand & Condense 🧮

1) Expand: $\log_2(8x^5)$ . What is the coefficient of $\log_2 x$ ? (e.g., in $\log_3(y^4) = 4\log_3 y$ , the coefficient is $4$ )

2) Condense: $\log 4 + \log 25$ . What is the single number inside the resulting $\log$ ? (e.g., $\log 3 + \log 7 = \log 21$ , so the number is )

3) Given $\log_b 2 = 0.5$ and $\log_b 3 = 0.8$ . Find , writing your answer as a decimal. (e.g., if and , then )

Rule Identification 🔽

Parameter	Effect	Example
$a$	Vertical stretch ($	a
$h$	Horizontal shift: right if $h>0$ , left if $h<0$	$h = 3$ : shift right 3
$k$	Vertical shift: up if $k>0$ , down if $k<0$	$k = -1$ : shift down 1
$b$	Base controls steepness: larger $b$ = less steep	$b = 10$ vs $b = 2$

📊 The Parent Function $y = \log_b x$

Key Points of $y = \log_2 x$ (parent)

$x$	$y = \log_2 x$
$\frac{1}{4}$

Features of Every Parent Log Function

Feature	Value
Domain	$(0, \infty)$
Range	$(-\infty, \infty)$
$x$ -intercept	$(1$ — always

🔄 Applying Transformations Step by Step

Worked Example

Graph $g(x) = -2\log_3(x + 1) + 4$ and identify all key features.

Start from the parent $y = \log_3 x$ and track the anchor point $(1, 0)$ :

Step	Transformation	Anchor Point	VA
Parent	$y = \log_3 x$	$(1, 0)$

Key Features of $g(x) = -2\log_3(x + 1) + 4$

Feature	Value
Domain	$(-1, \infty)$
Range	$(-\infty, \infty)$
VA	$x = -1$

Quick Rules for Domain & VA

$\boxed{\text{Domain of } \log_b(x - h): \quad x > h \quad \text{VA at } x = h}$

🎯 Finding the $x$ -Intercept Algebraically

Set $g(x) = 0$ and solve:

$-2\log_3(x + 1) + 4 = 0$

$\log_3(x + 1) = 2$

$x + 1 = 3^2 = 9$

$x = 8$

$x$ -intercept: $(8, 0)$ ✔

General Method

For $g(x) = a\log_b(x-h)+k$ , set $g = 0$ :

$\log_b(x - h) = -\frac{k}{a} \implies x = b^{-k/a} + h$

Finding the $y$ -Intercept

Set $x = 0$ : only exists if $0$ is in the domain (i.e., $h < 0$ ).

$g(0) = a\log_b(0 - h) + k = a\log_b(-h) + k$

Transformation Quiz 🎯

Graph Analysis 🧮

1) Find the $x$ -intercept of $f(x) = \log_2(x - 3) - 4$ . Set $f = 0$ , solve for $x$ . (e.g., for $\log_3(x-1) - 2 = 0$ : $\log_3(x-1) = 2$ , $x-1 = 9$ , $x = 10$ )

2) The domain of $g(x) = \ln(2x + 6)$ is $x >$ what value? (e.g., for $\ln(3x + 9)$ : set , so )

3) If $h(x) = 5\log(x) - 10$ , find $h(100)$ . (e.g., $3\log(1000) - 6 = 3(3) - 6 = 3$ )

Transformation Identification 🔽

⚠️ Always check solutions! Every candidate must make all original log arguments positive.

📝 Type 1: Single Log = Number

Worked Example 1

Solve $\log_3(2x + 1) = 4$

Step	Action	Result
1	Convert to exponential	$2x + 1 = 3^4 = 81$
2	Solve linear equation	$2x = 80$
3	Isolate $x$	$x = 40$
4	Check: $\log_3(2(40)+1)$	$= \log_3(81) = 4$ ✔

$\boxed{x = 40}$

Worked Example 2

Solve $\ln(x - 3) = 2$

$x - 3 = e^2 \implies x = e^2 + 3 \approx 10.389$

Check: $x - 3 = e^2 > 0$ ✔

📝 Type 2: Log = Log (One-to-One Property)

If $\log_b A = \log_b B$ , then $A = B$ (as long as both arguments are positive).

Worked Example 3

Solve $\log_2(x + 5) = \log_2(3x - 1)$

$x + 5 = 3x - 1$

$6 = 2x \implies x = 3$

Check: $\log_2(3+5) = \log_2(8) = 3$ and ✔

📝 Type 3: Multiple Logs — Condense First

Worked Example 4: Extraneous Solution Alert!

Solve $\log(x) + \log(x - 3) = 1$

Step	Action	Result
1	Product rule	$\log[x(x-3)] = 1$
2	Convert to exponential	$x(x-3) = 10^1 = 10$
3	Expand	$x^2 - 3x - 10 = 0$
4	Factor	$(x-5)(x+2) = 0$
5	Candidates	$x = 5$ or $x = -2$

Domain check — both original arguments must be positive:

$x = 5$ : $\log(5)$ ✔ and $\log(5-3) = \log(2)$ ✔ →

$\boxed{x = 5}$

⚠️ Why Extraneous Solutions Appear

When you condense logs, you may expand the domain. The product $x(x-3)$ can be positive even when the individual factors aren't both positive. Always check each original log argument separately.

Solving Strategies Quiz 🎯

Solve for $x$ 🧮

1) $\log_4(3x) = 2$ . Find $x$ . (e.g., $\log_3(2x) = 3$ : $2x = 27$ , $x = 13.5$ )

2) $\log(x) + \log(5) = 3$ . Find $x$ . (e.g., $\log(x) + \log(2) = 2$ : , , )

3) $2\ln(x) = \ln(25)$ . Find $x$ . (e.g., $2\ln(x) = \ln(9)$ : , , )

Strategy Selection 🔽

lo g_{b} x = \frac{lo g x}{lo g b} = \frac{ln x}{ln b}

Starting from $\log_b x = y$ :

$b^y = x \implies \ln(b^y) = \ln x \implies y \ln b = \ln x \implies y = \frac{\ln x}{\ln b}$

🧮 Evaluating with Change of Base

Example 1: $\log_7 50$

$\log_7 50 = \frac{\ln 50}{\ln 7} = \frac{3.912}{1.946} \approx 2.011$

Sanity check: $7^2 = 49 \approx 50$ ✔ (answer should be slightly above $2$ )

Example 2: $\log_3 100$

$\log_3 100 = \frac{\log 100}{\log 3} = \frac{2}{0.477} \approx 4.192$

Sanity check: $3^4 = 81$ and $3^5 = 243$ , so answer is between $4$ and ✔

Quick Reference for Common Calculations

Expression	Calculator Entry	Result
$\log_2 10$	$\ln(10)/\ln(2)$

🔗 Useful Relationships from Change of Base

Reciprocal Property

$\boxed{\log_b a = \frac{1}{\log_a b}}$

Example: $\log_2 8 = 3$ and $\log_8 2 = \frac{1}{3}$ . Product: ✔

Converting Between Bases

To convert $\log_a x$ into $\log_b x$ :

$\log_a x = \frac{\log_b x}{\log_b a}$

Change of Base in Equations

Solve $\log_2 x = \log_3 5$

Convert right side: $\log_3 5 = \frac{\ln 5}{\ln 3} \approx 1.465$

So $\log_2 x = 1.465 \implies x = 2^{1.465} \approx 2.760$

📊 Graphing Any Log with Change of Base

To graph $y = \log_b x$ on a calculator, enter:

$y = \frac{\ln x}{\ln b}$

Base Comparison Table

Base	$y = \log_b(10)$	Growth Rate	Steepness
$b = 2$

Key insight: Larger base = slower growth = flatter curve. All pass through $(1, 0)$ .

Change of Base Quiz 🎯

Calculator Practice 🧮

1) Evaluate $\log_2 32$ using change of base: $\frac{\log 32}{\log 2}$ . (e.g., $\log_3 27 = \frac{\log 27}{\log 3} = \frac{1.431}{0.477} = 3$ )

2) If $\log_4 7 \approx 1.404$ , find $\log_7 4$ to three decimal places. (e.g., if , then )

3) Evaluate $\log_9 27$ exactly. Hint: write both as powers of $3$ . (e.g., $\log_8 32$ : , , so answer is )

Base Fluency 🔽

dB = 10\log\left(\frac{I}{I_0}\right)

Key pattern: All involve $\log(\text{ratio})$ — they measure how many times larger one quantity is than a reference.

🧪 pH Scale

$\boxed{\text{pH} = -\log[H^+]}$

Worked Example 1

Orange juice has $[H^+] = 3.2 \times 10^{-4}$ M. Find its pH.

$\text{pH} = -\log(3.2 \times 10^{-4})$

$= -(\log 3.2 + \log 10^{-4})$

$= -(0.505 + (-4)) = -(0.505 - 4) = 3.495$

$\boxed{\text{pH} \approx 3.5}$

Worked Example 2 (Reverse)

A solution has pH $= 8.3$ . Find $[H^+]$ .

$-\log[H^+] = 8.3 \implies \log[H^+] = -8.3 \implies [H^+] = 10^{-8.3} \approx 5.01 \times 10^{-9}$ M

pH Comparison

Change	pH drops by	$[H^+]$ multiplied by
$1$ unit	$1$	$10$
units

A pH drop of $1$ means $10\times$ more acidic — that's the power of the log scale!

🔊 Decibel Scale

$\boxed{dB = 10\log\left(\frac{I}{I_0}\right)}$

where $I_0 = 10^{-12}\text{ W/m}^2$ (threshold of hearing).

Worked Example 3

A rock concert has intensity $I = 10^{-2}\text{ W/m}^2$ . Find the decibel level.

$dB = 10\log\left(\frac{10^{-2}}{10^{-12}}\right) = 10\log(10^{10}) = 10 \cdot 10 = 100\text{ dB}$

Common Sound Levels

Sound	Intensity (W/m²)	Decibels
Whisper	$10^{-10}$	$20$ dB
Conversation	$10^{-6}$

Comparing Two Sounds

If sound A is $10\text{ dB}$ louder than sound B, then A has $10\times$ the intensity.

$20\text{ dB}$ louder → $100\times$ intensity. $30\text{ dB}$ louder → $1{,}000\times$ intensity.

🌋 Richter Scale & Comparing Magnitudes

$\boxed{M = \log\left(\frac{A}{A_0}\right)}$

Comparing Two Earthquakes

How many times stronger is a magnitude $7$ earthquake than a magnitude $5$ ?

Each unit on the Richter scale represents $10\times$ the amplitude.

Difference: $7 - 5 = 2$ units → $10^2 = 100\times$ the amplitude

But energy scales by $\approx 31.6\times$ per unit:

$2$ units → $31.6^2 \approx 1{,}000\times$ the energy

Summary Table

Magnitude Difference	Amplitude Ratio	Energy Ratio
$1$	$10\times$	$\approx 31.6\times$
$2$	$100$

Log Models Quiz 🎯

Applied Calculations 🧮

1) Find the pH of a solution with $[H^+] = 10^{-9}$ M. (e.g., $[H^+] = 10^{-4}$ : pH $= -\log(10^{-4}) = 4$ )

2) A sound has intensity $I = 10^{-5}\text{ W/m}^2$ . Find its decibel level. Use $I_0 = 10^{-12}$ . (e.g., : )

3) How many times more intense is a $80\text{ dB}$ sound than a $50\text{ dB}$ sound? (e.g., $70\text{ dB}$ vs $40\text{ dB}$ : difference $30\text{ dB} = 10^3 = 1000\times$ )

Scale Identification 🔽

📋 Multi-Step Problem Walkthrough

A bacteria population is modeled by $P(t) = 500e^{0.2t}$ .

(a) When does the population reach $10{,}000$ ? (b) What is the doubling time? (c) Express the model in the form $P(t) = 500 \cdot b^t$ and find the percent growth rate.

Part (a): When $P = 10{,}000$ ?

$500e^{0.2t} = 10000$

$e^{0.2t} = 20$

$0.2t = \ln 20 \approx 2.996$

$t \approx 14.98 \approx 15$ time units

Part (b): Doubling time

$500e^{0.2t} = 1000 \implies e^{0.2t} = 2$

$t = \frac{\ln 2}{0.2} = \frac{0.693}{0.2} \approx 3.47$ time units

Part (c): Convert to $P = 500 \cdot b^t$

$e^{0.2t} = (e^{0.2})^t \approx (1.2214)^t$

So $b \approx 1.2214$ , meaning $\approx 22.14\%$ growth per time unit.

🗺️ Problem-Type Decision Map

If the Problem Says...	Strategy	First Move
"Evaluate $\log_b(\text{number})$ "	Definition	Rewrite as $b^? = \text{number}$
"Expand/simplify $\log(\text{expression})$ "	Properties (P2)	Apply product/quotient/power rules
"Graph $f(x) = a\log(x-h)+k$ "	Transformations (P3)	ID shifts, VA, anchor point
"Solve $\log(\text{stuff}) = \text{stuff}$ "	Equation solving (P4)	Condense logs → convert to exponential
"Compute $\log_b x$ to a decimal"	Change of base (P5)	$= \ln x / \ln b$
"Find the pH / dB / magnitude"	Modeling (P6)	Plug into the log-scale formula
"When does the population reach...?"	Log + exponential	Isolate exponential → take $\ln$

Connecting Logs to Exponentials

Almost every "when does it reach" problem follows this pattern:

$ab^t = c \implies b^t = \frac{c}{a} \implies t = \frac{\ln(c/a)}{\ln b}$

Synthesis Quiz 🎯

Multi-Skill Drill 🧮

1) An investment grows as $A = 2000e^{0.06t}$ . How many years to reach $\$ 6{,}000 $? Round to one decimal. (e.g.,$ 1000e^{0.05t} = 3000 $:$ t = \frac{\ln 3}{0.05} = \frac{1.099}{0.05} = 22.0$ years)

2) Condense: $2\log x - \frac{1}{2}\log y$ . Write the coefficient on $y$ when the expression equals . What is as a fraction? (e.g., , so )

3) An earthquake of magnitude $4$ vs magnitude $7$ : the stronger one has how many times the amplitude? (e.g., magnitude $3$ vs $5$ : difference $2$ , ratio $= 10^2 = 100$ )

Concept Connection 🔽

Final Exit Quiz — Logarithmic Functions ✅

(- \infty, \infty)

\log_b 10 = 1.2 + 0.5 = 1.7

3 lo g (1000) - 6 = 3 (3) - 6 = 3

\log_2(3(3)-1) = \log_2(8) = 3

x = -2

\log(-2)

❌ → extraneous, reject

3 \times \frac{1}{3} = 1

lo g_{3} 5 \approx 1.465

\log_5 3 = 1/1.465 \approx 0.683

dB = 10\log(10^{-8}/10^{-12}) = 10 \cdot 4 = 40

\log\left(\frac{x^a}{y^b}\right)

3\log x - 2\log y = \log(x^3/y^2)

Logarithmic Functions and Equations

Logarithmic Functions and Equations - Complete Interactive Lesson

Part 1: Introduction to Logarithms

📖 Logarithmic Functions — Definition & Inverse Relation

The Three Standard Bases

Quick Conversion Examples

🔄 Logs and Exponentials Are Inverses

📝 Worked Example: Converting & Evaluating

Part 2: Properties of Logarithms

📐 Logarithmic Functions — Core Log Properties

The Three Fundamental Properties

Part 3: Solving Logarithmic Equations

📈 Logarithmic Functions — Transformations & Graphs

Part 4: Change of Base

🔍 Logarithmic Functions — Solving Log Equations

The Two Core Strategies

Part 5: Logarithmic Models

🔀 Logarithmic Functions — Change of Base & Calculator Fluency

Part 6: Problem-Solving Workshop

🌍 Logarithmic Functions — Modeling with Logs

Real-World Log Scales

Part 7: Review & Applications

🏆 Logarithmic Functions — Full Synthesis

Your Complete Log Toolkit

Graphical Connection

Pattern for Evaluating Logs Mentally

Key Values to Memorize

🔓 Expanding Logarithmic Expressions

Worked Example 1

Order of Operations for Expanding

🔒 Condensing Logarithmic Expressions

Worked Example 2

Why Condensing Matters

🚫 Common Errors to Avoid

Quick Memory Aid

Parameter Effects

📊 The Parent Function y=log⁡bxy = \log_b xy=logb​x

Key Points of y=log⁡2xy = \log_2 xy=log2​x (parent)

Features of Every Parent Log Function

🔄 Applying Transformations Step by Step

Worked Example

Key Features of g(x)=−2log⁡3(x+1)+4g(x) = -2\log_3(x + 1) + 4g(x)=−2log3​(x+1)+4

Quick Rules for Domain & VA

🎯 Finding the xxx-Intercept Algebraically

General Method

Finding the yyy-Intercept

📝 Type 1: Single Log = Number

Worked Example 1

Worked Example 2

📝 Type 2: Log = Log (One-to-One Property)

Worked Example 3

📝 Type 3: Multiple Logs — Condense First

Worked Example 4: Extraneous Solution Alert!

⚠️ Why Extraneous Solutions Appear

Why It Works

🧮 Evaluating with Change of Base

Example 1: log⁡750\log_7 50log7​50

Example 2: log⁡3100\log_3 100log3​100

Quick Reference for Common Calculations

🔗 Useful Relationships from Change of Base

Reciprocal Property

Converting Between Bases

Change of Base in Equations

📊 Graphing Any Log with Change of Base

Base Comparison Table

🧪 pH Scale

Worked Example 1

Worked Example 2 (Reverse)

pH Comparison

🔊 Decibel Scale

Worked Example 3

Common Sound Levels

Comparing Two Sounds

🌋 Richter Scale & Comparing Magnitudes

Comparing Two Earthquakes

Summary Table

📋 Multi-Step Problem Walkthrough

Part (a): When P=10,000P = 10{,}000P=10,000?

Part (b): Doubling time

Part (c): Convert to P=500⋅btP = 500 \cdot b^tP=500⋅bt

📊 The Parent Function $y = \log_b x$

Key Points of $y = \log_2 x$ (parent)

Key Features of $g(x) = -2\log_3(x + 1) + 4$

🎯 Finding the $x$ -Intercept Algebraically

Finding the $y$ -Intercept

Example 1: $\log_7 50$

Example 2: $\log_3 100$

Part (a): When $P = 10{,}000$ ?

Part (c): Convert to $P = 500 \cdot b^t$