When the Base Is Not Written

Idea

A logarithm written with no base at all, $\log 50$ , is not missing anything. The base is $10$ , left out because it is the one people write most often.

We count in tens, so $10$ is the base that makes place value readable. This is the common logarithm.

Rule

$\log x = \log_{10} x$

$\log 10^n = n$

$10^{\log x} = x$

How it is used

1. Put the ten back

Nothing new happens here. Read the missing base as a $10$ and every earlier lesson applies word for word.

Written	Means	Asks
$\log 100$	$\log_{10} 100$	$10$ to what power gives $100$
$\log 0.001$	$\log_{10} 0.001$	$10$ to what power gives $0.001$
$\log x$	$\log_{10} x$	$10$ to what power gives $x$

Only $\log$ hides its base. $\log_2 8$ still needs its $2$ written.

2. Every power of ten is free

The exponent is the answer. Read it straight off.

$x$	$1000$	$100$	$10$	$1$	$0.1$	$0.01$
as a power	$10^3$	$10^2$	$10^1$	$10^0$	$10^{-1}$	$10^{-2}$
$\log x$	$3$	$2$	$1$	$0$	$-1$	$-2$

A number below $1$ gives a negative logarithm, because it needs a negative power of ten.

3. The four rules, with the ten hidden

\log 1 = 0

\log 10 = 1

\log 10^n = n

10^{\log x} = x

These are the four rules from the last lesson with $a = 10$ . The last one is the one students forget: a $10$ raised to a common logarithm cancels it and hands back the argument, whatever the argument is.

$10^{\log 7} = 7$

$10^{\log(x + 4)} = x + 4$

4. Everything else sits between two integers

Most numbers are not powers of ten. Trap such a number between the two powers of ten either side, and its logarithm is trapped between those two exponents.

Trapped between	So $\log x$ is between
$10 < 50 < 100$	$1$ and $2$
$100 < 700 < 1000$	$2$ and $3$
$0.01 < 0.04 < 0.1$	$-2$ and $-1$

This is the sanity check to run on every calculator answer. If $\log 50$ comes back as $3.9$ , something was typed wrong.

Exponential and logarithm

Try this. Compare bases 2 and 0.5: growth becomes decay. Move the input to 1; the logarithm is zero for either base.

BasePositive input x0.5

Base 3 · Blue: y = 3ˣ · Green: y = log_b(x) · Dashed: y = x. At x = 0.5, y = -0.631.

Worked example

Evaluate $\log 1000 + \log 0.01 - \log 1 + 10^{\log 7}$ .

Remember

$\log$ with no base written means base $10$ $\log 10^n = n$ , so read the exponent straight off $10^{\log x} = x$ , the two cancel

Write each power of ten in index form.

\log 10^3 + \log 10^{-2} - \log 1 + 10^{\log 7}

The first two are exponents read off directly.

3 + (-2) - \log 1 + 10^{\log 7}

$\log 1 = 0$ for every base, this one included.

3 - 2 - 0 + 10^{\log 7}

The last term cancels to its argument.

3 - 2 - 0 + 7 = 8

Explore

Nothing about base $10$ is special except the number itself. Drag the base slider and watch what refuses to change: the curve always passes through $(1, 0)$ , always climbs, always hugs the $y$ -axis on the left.

What does change is steepness. A larger base makes the climb gentler, and the slider only reaches $4$ , so picture base $10$ as the same green curve flattened further still. That flatness is the whole story of the common logarithm: it takes a jump all the way from $x = 10$ to $x = 100$ to raise the height by one.