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: $\log_3(81)$

💡 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: $4$

2Problem 2medium

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

3Problem 3hard

❓ 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: $x = 5$

🎴

Practice with Flashcards

Review key concepts with our flashcard system

📖

Browse All Topics

Explore other calculus topics