Common log (log₁₀) and natural log (ln x): Question bank

Definition Ground rules used on this page (read first)

The number $e$ . $e \approx 2.71828$ is a fixed constant — the base of the natural logarithm. It is defined by $e = lim_{n \to \infty} (1 + \frac{1}{n})^{n}$ (see Exponential functions e^x and 10^x). So " $ln x$ " just means " $lo g_{e} x$ ", asking " $e$ to what power gives $x$ ?"
Valid bases. A logarithm $lo g_{b} x$ requires the base $b > 0$ and $b \neq = 1$ (if $b = 1$ , every power of $1$ is $1$ , so " $1$ to what power = $x$ ?" has no answer), and the argument $x > 0$ . Common log uses $b = 10$ ; natural log uses $b = e$ .
Change of base (invoked several times below): $lo g_{b} a = \frac{l n a}{l n b} = \frac{l o g _{10} a}{l o g _{10} b} .$ Why it holds: let $y = lo g_{b} a$ , so $b^{y} = a$ . Take $ln$ of both sides: $y ln b = ln a$ , hence $y = \frac{l n a}{l n b}$ . This is what lets us rewrite any base in terms of $ln$ or $lo g_{10}$ .
Base bigger or smaller than 1. For $b > 1$ (like $10$ and $e$ ), $lo g_{b} x$ is increasing. For $0 < b < 1$ it is decreasing — a rarely-seen but valid edge case, and the opposite direction of everything below.

True or false — justify

lo g_{10} (1000) = 3

because there are three zeros.

True in spirit, but the real reason is

1000 = 1 0^{3}

; the log counts powers of ten, not literally zeros. For

2000

the "count the zeros" trick fails (

lo g_{10} 2000 \approx 3.30

), so trust the exponent, not the zeros.

For every real number

x

ln (e^{x}) = x

True.

ln

is the exact inverse of

e^{(\cdot)}

, so it undoes the exponential for all real

x

, including negatives and zero — because

e^{x}

can be any positive value.

For every real number

x

e^{l n x} = x

Only true for

x > 0

. Here

ln x

must exist first, and

ln

is defined only on positive

x

, so the identity is restricted to the domain of

ln

ln x

grows without bound, so it must eventually exceed any straight line

y = m x

False.

ln x

grows slower than any positive-slope line; for large

x

the line wins.

ln

increases forever but ever more slowly (its slope

1/ x \to 0

lo g_{10} x

and

ln x

are just vertically stretched copies of each other.

True — they differ by a constant factor:

ln x = ln 10 \cdot lo g_{10} x \approx 2.303 lo g_{10} x

. Same shape, same roots (both hit zero at

x = 1

), one scaled version of the other.

lo g_{10}

of a number between

0

and

1

is negative.

True. Such numbers are

1 0^{(negative)}

, e.g.

0.01 = 1 0^{- 2}

, so the log — asking "10 to what power?" — returns a negative exponent.

ln 1 = 0

and

lo g_{10} 1 = 0

True for both and for every valid base. Any base to the power

0

1

, so "what power gives 1?" is always

0

. This is the one input where all logs agree.

The graph of

ln x

crosses the

y

-axis.

False. Its domain is

x > 0

, so there is no point at

x = 0

. As

x \to 0^{+}

the curve plunges to

- \infty

but never touches the

y

-axis — the

y

-axis is a vertical asymptote.

A logarithm base

b

is always an increasing function.

False. It is increasing only when

b > 1

(as with

10

and

e

). For

0 < b < 1

the function is decreasing, because raising a base below

1

to a larger power gives a smaller number.

Spot the error

lo g (a + b) = lo g a + lo g b

Wrong. Logs turn products into sums, not sums into sums:

lo g (ab) = lo g a + lo g b

. Test:

lo g_{10} (10 + 10) = lo g_{10} 20 \approx 1.30

, but

lo g_{10} 10 + lo g_{10} 10 = 2

\frac{lo g 8}{lo g 2} = lo g 8 - lo g 2 = lo g 4

Wrong — a quotient of logs is not the log of a quotient. By change of base

\frac{lo g 8}{lo g 2} = lo g_{2} 8 = 3

, whereas

lo g (8/2) = lo g 4 \approx 0.60

. Division of logs and log of division are different operations.

lo g (- 5)

is just a negative log, so it equals

- lo g 5

Wrong.

lo g (- 5)

is undefined in the reals: no power of

10

gives a negative number since

1 0^{y} > 0

always. Don't confuse a negative input (illegal) with a negative output (fine, as in

lo g 0.5

(ln x)^{2} = 2 ln x

Wrong — that misapplies the power law. The law

ln (x^{2}) = 2 ln x

needs the square inside the log.

(ln x)^{2}

is the log squared, an entirely different quantity.

"To solve

1 0^{x} = 7

I take

lo g_{10}

and get

x = lo g_{10} 10 \cdot lo g_{10} 7

Wrong. Taking

lo g_{10}

of both sides gives

x lo g_{10} 10 = lo g_{10} 7

, i.e.

x \cdot 1 = lo g_{10} 7

, so

x = lo g_{10} 7

. The exponent comes down as a coefficient; you don't multiply two logs.

ln

and

lo g

are the same calculator button, pick either."

Wrong. log = base

10

, ln = base

e

. They differ by a factor of

ln 10 \approx 2.303

, so using the wrong one scales the answer by about

2.3 \times

"Since

ln 20 \approx 3

, then

lo g_{10} 20 \approx 3

too."

Wrong — that ignores the base.

ln 20 \approx 3.00

but

lo g_{10} 20 \approx 1.30

; they differ by the

ln 10 \approx 2.303

factor. Same number, different rulers.

Why questions

Why is the domain of

lo g x

restricted to

x > 0

Because

lo g x

asks "base to what power gives

x

?", and any positive base raised to any real power is always positive. No real exponent can produce

0

or a negative, so those inputs have no answer.

Why do we require the base

b

to satisfy

b > 0

and

b \neq = 1

A negative or zero base can't be raised to arbitrary real powers consistently, and

b = 1

gives

1^{y} = 1

for every

y

, so "

1

to what power =

x

?" has no unique answer unless

x = 1

. Only

b > 0, b \neq = 1

gives a well-defined inverse.

Why does base

e

appear everywhere in calculus rather than base

10

Because

\frac{d}{d x} e^{x} = e^{x}

(it copies its own slope), which makes

\frac{d}{d x} ln x = \frac{1}{x}

— perfectly clean. Any other base drags along a stray factor of

ln (base)

Why does

lo g_{10}

measure "orders of magnitude"?

Because each whole-number increase in

lo g_{10} x

means

x

was multiplied by

10

. So the log counts how many powers of ten a number spans — exactly what "order of magnitude" means.

Why is

lo g_{10} e

the reciprocal of

ln 10

By change of base,

lo g_{10} e = \frac{l n e}{l n 10} = \frac{1}{l n 10}

since

ln e = 1

. In general

lo g_{a} b

and

lo g_{b} a

are always reciprocals.

Why does the minus sign appear in

pH = - lo g_{10} [H^{+}]

Because

[H^{+}]

is a tiny number like

1 0^{- 4}

, giving a negative log. The minus flips it to a friendly positive value, and each pH unit still represents a factor of

10

in acidity.

Why can we take

ln

of both sides of an equation and keep it valid?

Because

ln

is a one-to-one function on positives: if two positive quantities are equal, their logs are equal and vice versa. This preserves the equation while letting the power law pull exponents down.

Why does

(1 + \frac{1}{n})^{n}

settle on

e

instead of blowing up to infinity?

Take the log:

ln (1 + \frac{1}{n})^{n} = n ln (1 + \frac{1}{n})

. Writing

h = \frac{1}{n} \to 0

, this is

\frac{l n ( 1 + h )}{h}

, which by l'Hôpital (or the known limit) tends to

\frac{1/ ( 1 + h )}{1} \to 1

. So the log of the expression

\to 1

, meaning the expression itself

\to e^{1} = e

— a finite value, not infinity.

Edge cases

What is

lo g_{10} (0)

Undefined. There is no power of

10

equal to

0

; as

x \to 0^{+}

lo g_{10} x \to - \infty

, so it approaches negative infinity but never attains a value at

0

What is

ln (e)

, and what is

ln (1/ e)

ln e = 1

(one factor of

e

), and

ln (1/ e) = ln (e^{- 1}) = - 1

. The reciprocal input flips the sign of the log.

lo g_{10} (x)

ever equal to

ln (x)

for some

x

Only at

x = 1

, where both equal

0

. Everywhere else the constant factor

ln 10 \approx 2.303

separates them; they share just that single crossing point.

What happens to

ln x

x \to 0^{+}

versus

x \to \infty

x \to 0^{+}

ln x \to - \infty

(steeply down, vertical asymptote at the

y

-axis). As

x \to \infty

ln x \to + \infty

but ever more slowly, since its slope

1/ x

shrinks toward

0

Can

lo g_{b} x

be negative even when

x

is positive?

Yes — whenever

0 < x < 1

(with base

b > 1

). For example

lo g_{10} (0.1) = - 1

. A positive input smaller than

1

needs a negative exponent.

How does

lo g_{b} x

behave when

0 < b < 1

, say

b = \frac{1}{2}

It is a decreasing function: bigger

x

gives smaller

lo g_{b} x

. For instance

lo g_{1/2} (8) = - 3

since

(\frac{1}{2})^{- 3} = 8

. This flips the usual "growth" intuition from base

10

and base

e

Is there any base

b

for which

lo g_{b} x

is defined at

x = 0

No. For any valid base,

b^{y} > 0

for all real

y

, so

0

is never in the range of

b^{y}

and never in the domain of

lo g_{b}

What is

ln 10 \cdot lo g_{10} e

, and why?

It equals exactly

1

. Since

lo g_{10} e = \frac{1}{l n 10}

, the product is

ln 10 \cdot \frac{1}{l n 10} = 1

— a reciprocal pair.

Connections

Laws of Logarithms (product, quotient, power) — the rules these traps most often abuse.
Solving exponential equations — where the "take log of both sides" moves live.
Exponential functions e^x and 10^x — the inverses that fix the domain to $x > 0$ , and where $e$ is defined.
Differentiation of ln x and e^x — why base $e$ is the calculus-natural choice.
Scientific notation and orders of magnitude — the home of "orders of magnitude" reasoning.