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

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

Common Logarithms

1. Common logarithm (base 10): $\log(x) = \log_{10}(x)$
2. Natural logarithm (base $e$): $\ln(x) = \log_e(x)$

Converting Between Forms

Properties of Logarithms

Product Rule

$\log_b(MN) = \log_b(M) + \log_b(N)$

The log of a product is the sum of the logs.

Quotient Rule

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

The log of a quotient is the difference of the logs.

Power Rule

$\log_b(M^p) = p \cdot \log_b(M)$

The log of a power is the exponent times the log.

Change of Base Formula

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)} = \frac{\ln(x)}{\ln(b)}$

Useful for calculating logs with different bases on a calculator.

Special Logarithm Values

1. $\log_b(1) = 0$ because $b^0 = 1$
2. $\log_b(b) = 1$ because $b^1 = b$
3. $\log_b(b^x) = x$ (inverse property)
4. $b^{\log_b(x)} = x$ (inverse property)

Logarithmic Functions

The function $f(x) = \log_b(x)$ has these properties:

• Domain: $(0, \infty)$ (only positive numbers)
• Range: $(-\infty, \infty)$ (all real numbers)
• Vertical Asymptote: $x = 0$ (the y-axis)
• x-intercept: $(1, 0)$ since $\log_b(1) = 0$
• Always increasing if $b > 1$
• Always decreasing if $0 < b < 1$

Solving Logarithmic Equations

Strategy 1: Convert to exponential form

Strategy 2: Use logarithm properties to combine/simplify

Strategy 3: Check your answers (domain restrictions!)

Important Note

⚠️ You cannot take the log of a negative number or zero!

The domain of $\log_b(x)$ requires $x > 0$.

Therefore: $\log_2 32 = 5$

Part (b): $\log_5 \frac{1}{25}$ means "5 to what power equals $\frac{1}{25}$?"

$\frac{1}{25} = \frac{1}{5^2} = 5^{-2}$

Therefore: $\log_5 \frac{1}{25} = -2$

Part (c): $\ln e^7$ means $\log_e e^7$

By the property $\log_b b^x = x$:

$\ln e^7 = 7$

Step 2: Apply the product rule to the first term.

$= \log_2(8) + \log_2(x^3) - \log_2(y^2)$

Step 3: Apply the power rule.

$= \log_2(8) + 3\log_2(x) - 2\log_2(y)$

Step 4: Simplify $\log_2(8)$.

Since $2^3 = 8$, we have $\log_2(8) = 3$

$= 3 + 3\log_2(x) - 2\log_2(y)$

Answer: $3 + 3\log_2(x) - 2\log_2(y)$

Step 1: Apply quotient rule:

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

Step 2: Apply product rule to the first term:

$= \log_3(x^4) + \log_3(\sqrt{y}) - \log_3(z^2)$

Step 3: Rewrite $\sqrt{y} = y^{1/2}$ and apply power rule:

$= 4\log_3 x + \frac{1}{2}\log_3 y - 2\log_3 z$

Final answer: $4\log_3 x + \frac{1}{2}\log_3 y - 2\log_3 z$

Step 2: Evaluate.

$x = 4^3 = 64$

Step 3: Check (optional but recommended).

$\log_4(64) = \log_4(4^3) = 3$ ✓

Answer: $x = 64$