Logarithm Properties

The Three Main Properties

These come directly from exponent laws.

$\log_b(xy) = \log_b(x) + \log_b(y)$

Log of a product = sum of the logs.

Why it works:

If $b^m = x$ and $b^n = y$ , then:

$xy = b^m \cdot b^n = b^{m+n}$

So $\log_b(xy) = m + n = \log_b(x) + \log_b(y)$ .

Example:

$\log(6) = \log(2 \times 3) = \log(2) + \log(3)$

$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

Log of a quotient = difference of the logs.

Why it works:

If $b^m = x$ and $b^n = y$ , then:

$\frac{x}{y} = \frac{b^m}{b^n} = b^{m-n}$

So $\log_b\left(\frac{x}{y}\right) = m - n = \log_b(x) - \log_b(y)$ .

Example:

\begin{aligned} \log(5) &= \log\left(\frac{10}{2}\right) \\ &= \log(10) - \log(2) \\ &= 1 - \log(2) \end{aligned}

$\log_b(x^n) = n \cdot \log_b(x)$

Log of a power = exponent times the log.

Why it works:

If $b^m = x$ , then $m = \log_b(x)$ , and:

\begin{aligned} x^n &= (b^m)^n \\ &= b^{mn} \end{aligned}

So $\log_b(x^n) = mn$ .

Substitute $m = \log_b(x)$ :

$\log_b(x^n) = n \cdot \log_b(x)$

Example:

$\log(8) = \log(2^3) = 3 \cdot \log(2)$

Property	Rule
Product	$\log_b(xy) = \log_b(x) + \log_b(y)$
Quotient	$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
Power	$\log_b(x^n) = n \cdot \log_b(x)$

$\log_b(1) = 0$

Because $b^0 = 1$ .

$\log_b(b) = 1$

Because $b^1 = b$ .

To convert between bases:

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

Most useful form:

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

Example: Calculate $\log_2(10)$

\begin{aligned} \log_2(10) &= \frac{\log(10)}{\log(2)} \\[0.5em] &= \frac{1}{0.301} \\[0.5em] &\approx 3.32 \end{aligned}