Logarithms

Everything for this chapter, in one place. Throughout, bases are positive and not 1, and arguments are positive.

What it means

The laws — they are the exponent laws, re-read


Product	$\log(mn) = \log m + \log n$
Quotient	$\log\dfrac{m}{n} = \log m - \log n$
Power	$\log(m^p) = p\log m$
A sum of logs	is the log of a product — always combine before solving

Changing the base — the engine of the chapter


General	$\log_b a = \dfrac{\log_c a}{\log_c b}$
Flip	$\log_b a = \dfrac{1}{\log_a b}$
Powers in both places	$\log_{b^p}(a^q) = \dfrac{q}{p}\log_b a$
Swap the exponents	$a^{\log_b c} = c^{\log_b a}$
A chain telescopes	$\log_a b \cdot \log_b c \cdot \log_c d = \log_a d$

When several powers are equal


If $a^x = b^y = c^z = k$	$\dfrac{1}{x} = \log_k a$ , $\dfrac{1}{y} = \log_k b$ , $\dfrac{1}{z} = \log_k c$
Which makes	sums and differences of $\dfrac{1}{x}$ into logs of products and quotients

Sign and direction — where the traps live


$\log_b a > 0$	when $a$ and $b$ are on the same side of 1
$\log_b a < 0$	when they are on opposite sides
Base $> 1$	the log increases, so inequalities keep their direction
Base between 0 and 1	the log decreases, so inequalities flip
Domain	the argument must be strictly positive, and the base positive and $\neq 1$

Useful values


$\log_{10} 2 \approx 0.301$	$\log_{10} 3 \approx 0.477$
$\log_2 3 \approx 1.585$	$\log_2 10 \approx 3.322$

Formula sheet