Logarithmic Functions

📖 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 $x$."

The Three Standard Bases

Quick Conversion Examples

🔄 Logs and Exponentials Are Inverses

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

Inverse Properties — they undo each other:

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

Graphical Connection

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

📝 Worked Example: Converting & Evaluating

Evaluate $\log_4 64$ without a calculator.

Step 1: Rewrite as an equation: $\log_4 64 = x$ means $4^x = 64$

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

Key Values to Memorize

| $\log_b 1 = 0$ | Because $b^0 = 1$ for any base | | $\log_b b = 1$ | Because $b^1 = b$ | | $\log_b b^n = n$ | Inverse property |

Concept Check 🎯

Evaluate These Logs 🧮

1) $\log_5 125 =$ ? (e.g., $\log_2 64$: since $2^6 = 64$, the answer is $6$)

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 $-2$)

Log Fundamentals 🔽

Exit Quiz ✅

📐 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

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

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

Order of Operations for Expanding

1. Quotient rule first (handle the fraction)
2. Product rule next (break up any remaining products)
3. 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 $3\ln a - \frac{1}{2}\ln b + \ln c$

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

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

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 $21$)

3) Given $\log_b 2 = 0.5$ and $\log_b 3 = 0.8$. Find $\log_b 12$, writing your answer as a decimal. (e.g., if $\log_b 5 = 1.2$ and $\log_b 2 = 0.5$, then $\log_b 10 = 1.2 + 0.5 = 1.7$)

Rule Identification 🔽

Exit Quiz ✅

📈 Logarithmic Functions — Transformations & Graphs

Part 3 of 7

The general transformed logarithmic function:

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

Parameter Effects

📊 The Parent Function $y = \log_b x$

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

Features of Every Parent Log Function

🔄 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 $(1, 0)$:

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

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 $3x + 9 > 0$, so $x > -3$)

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

Transformation Identification 🔽

Exit Quiz ✅

🔍 Logarithmic Functions — Solving Log Equations

Part 4 of 7

The Two Core Strategies

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

📝 Type 1: Single Log = Number

Worked Example 1

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

$\boxed{x = 40}$

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 $\log_2(3(3)-1) = \log_2(8) = 3$ ✔

📝 Type 3: Multiple Logs — Condense First

Worked Example 4: Extraneous Solution Alert!

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

Domain check — both original arguments must be positive:

• $x = 5$: $\log(5)$ ✔ and $\log(5-3) = \log(2)$ ✔ → valid
• $x = -2$: $\log(-2)$ ❌ → extraneous, reject

$\boxed{x = 5}$

⚠️ 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., $\log(x) + \log(2) = 2$: $\log(2x) = 2$, $2x = 100$, $x = 50$)

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

Strategy Selection 🔽

Exit Quiz ✅

🔀 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:

$\boxed{\log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}}$

Why It Works

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 $5$ ✔

Quick Reference for Common Calculations

🔗 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: $3 \times \frac{1}{3} = 1$ ✔

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

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 $\log_3 5 \approx 1.465$, then $\log_5 3 = 1/1.465 \approx 0.683$)

3) Evaluate $\log_9 27$ exactly. Hint: write both as powers of $3$. (e.g., $\log_8 32$: $8 = 2^3$, $32 = 2^5$, so answer is $\frac{5}{3}$)

Base Fluency 🔽

Exit Quiz ✅

🌍 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

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

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

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

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., $I = 10^{-8}$: $dB = 10\log(10^{-8}/10^{-12}) = 10 \cdot 4 = 40$)

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 🔽

Exit Quiz ✅

🏆 Logarithmic Functions — Full Synthesis

Part 7 of 7 — Putting It All Together

Your Complete Log Toolkit

📋 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

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 $\log\left(\frac{x^a}{y^b}\right)$. What is $b$ as a fraction? (e.g., $3\log x - 2\log y = \log(x^3/y^2)$, so $b = 2$)

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 ✅