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Math notes/Pre-Calculus/Logarithms

When the Base Is Not Written — Cheat sheet

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The move

No base written means base $10$ .

\log x = \log_{10} x

Read the exponent off

$x$	$10^n$	$\log x$
$10000$	$10^4$	$4$
$1000$	$10^3$	$3$
$100$	$10^2$	$2$
$10$	$10^1$	$1$
$1$	$10^0$	$0$
$0.1$	$10^{-1}$	$-1$
$0.01$	$10^{-2}$	$-2$
$0.001$	$10^{-3}$	$-3$

The four rules with the ten hidden

Rule	Example
$\log 1 = 0$	$\log 1 = 0$
$\log 10 = 1$	$\log 10 = 1$
$\log 10^n = n$	$\log 10^6 = 6$
$10^{\log x} = x$	$10^{\log 7} = 7$

Sanity check any answer

Trap the number between the powers of ten either side.

Because	$\log x$ lies between
$10 < 50 < 100$	$1$ and $2$
$100 < 700 < 1000$	$2$ and $3$
$1 < 5 < 10$	$0$ and $1$
$0.01 < 0.04 < 0.1$	$-2$ and $-1$

Watch out

Only $\log$ hides a base. $\log_2 8$ keeps its $2$ .
$x$ must still be positive: $\log 0$ and $\log(-5)$ have no value.
A number under $1$ gives a negative answer, never an undefined one.