Logarithmic Functions

Understanding and evaluating logarithms

Logarithmic Functions

Definition

A logarithm is the inverse of an exponential function.

$\log_b(x) = y \quad \text{means} \quad b^y = x$

Read as: "log base $b$ of $x$ equals $y$ "

Example: $\log_2(8) = 3$ because $2^3 = 8$

Common Logarithms

Common log: $\log(x)$ means $\log_{10}(x)$

Natural log: $\ln(x)$ means $\log_e(x)$ where $e \approx 2.718$

Properties of Logarithms

Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$

Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$

Power Rule: $\log_b(M^p) = p \cdot \log_b(M)$

Change of Base: $\log_b(x) = \frac{\log(x)}{\log(b)}$

Special Values

$\log_b(1) = 0$ (because $b^0 = 1$ )
$\log_b(b) = 1$ (because $b^1 = b$ )
$\log_b(b^x) = x$
$b^{\log_b(x)} = x$

📚 Practice Problems

1Problem 1easy

❓ Question:

Evaluate: log₂ 8

💡 Show Solution

Step 1: Understand the question: log₂ 8 means "2 to what power equals 8?"

Step 2: Find the power: 2¹ = 2 2² = 4 2³ = 8

Step 3: Answer: Since 2³ = 8, we have log₂ 8 = 3

Answer: 3

2Problem 2easy

❓ Question:

Evaluate: $\log_3(81)$

💡 Show Solution

We need to find: $\log_3(81) = ?$

This asks: "3 to what power equals 81?"

$3^? = 81$

Since $3^4 = 81$ : $\log_3(81) = 4$

Answer: $4$

3Problem 3easy

❓ Question:

Evaluate: log₁₀ 1000

💡 Show Solution

Step 1: Rewrite as an exponential equation: log₁₀ 1000 = x means 10ˣ = 1000

Step 2: Express 1000 as a power of 10: 1000 = 10³

Step 3: Therefore: log₁₀ 1000 = 3

Step 4: Note: log₁₀ is called the "common logarithm" Often written as just "log" without the base

Answer: 3

4Problem 4medium

❓ Question:

Expand using log properties: $\log_5(\frac{x^3y}{z^2})$

💡 Show Solution

Use quotient, product, and power rules:

Step 1: Apply quotient rule $\log_5\left(\frac{x^3y}{z^2}\right) = \log_5(x^3y) - \log_5(z^2)$

Step 2: Apply product rule to first term $= \log_5(x^3) + \log_5(y) - \log_5(z^2)$

Step 3: Apply power rule $= 3\log_5(x) + \log_5(y) - 2\log_5(z)$

Answer: $3\log_5(x) + \log_5(y) - 2\log_5(z)$

5Problem 5medium

❓ Question:

Convert to logarithmic form: 5³ = 125

💡 Show Solution

Step 1: Recall the relationship: bˣ = y is equivalent to logᵦ y = x

Step 2: Identify the parts: Base (b) = 5 Exponent (x) = 3 Result (y) = 125

Step 3: Write in logarithmic form: log₅ 125 = 3

Step 4: Verify: "5 to what power equals 125?" 5³ = 125 ✓

Answer: log₅ 125 = 3

6Problem 6medium

❓ Question:

Simplify using logarithm properties: log₃ 27 + log₃ 9

💡 Show Solution

Step 1: Use the product rule: logᵦ m + logᵦ n = logᵦ(mn)

Step 2: Apply the rule: log₃ 27 + log₃ 9 = log₃(27 · 9) = log₃ 243

Step 3: Evaluate log₃ 243: What power of 3 equals 243? 3¹ = 3 3² = 9 3³ = 27 3⁴ = 81 3⁵ = 243

Step 4: Therefore: log₃ 243 = 5

Alternative - evaluate first: log₃ 27 = 3 (since 3³ = 27) log₃ 9 = 2 (since 3² = 9) 3 + 2 = 5 ✓

Answer: 5

7Problem 7hard

❓ Question:

Solve: $\log_2(x + 3) + \log_2(x - 3) = 4$

💡 Show Solution

Step 1: Use product rule (combine logs) $\log_2[(x + 3)(x - 3)] = 4$

Step 2: Convert to exponential form $(x + 3)(x - 3) = 2^4 = 16$

Step 3: Simplify left side (difference of squares) $x^2 - 9 = 16$

Step 4: Solve for x $x^2 = 25$ $x = \pm 5$

Step 5: Check both solutions

$x = 5$ : $\log_2(8) + \log_2(2) = 3 + 1 = 4$ ✓
$x = -5$ : $\log_2(-2) + \log_2(-8)$ ✗ (negative logs undefined)

Answer: $x = 5$

8Problem 8hard

❓ Question:

Expand using logarithm properties: log₂(8x³/y²)

💡 Show Solution

Step 1: Apply the quotient rule: logᵦ(m/n) = logᵦ m - logᵦ n

log₂(8x³/y²) = log₂(8x³) - log₂(y²)

Step 2: Apply the product rule to first term: logᵦ(mn) = logᵦ m + logᵦ n

log₂(8x³) = log₂ 8 + log₂ x³

Step 3: Apply the power rule: logᵦ(mⁿ) = n logᵦ m

log₂ x³ = 3 log₂ x log₂ y² = 2 log₂ y

Step 4: Combine all parts: log₂(8x³/y²) = log₂ 8 + 3 log₂ x - 2 log₂ y

Step 5: Simplify log₂ 8: log₂ 8 = 3 (since 2³ = 8)

Step 6: Final answer: 3 + 3 log₂ x - 2 log₂ y

Answer: 3 + 3 log₂ x - 2 log₂ y

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