# Box 6.0: Common Exponential and Log Identities

*
© 1999 by Karl Hahn*

*Adding the Exponents:*
If `b` is any positive real number then

b^{x} b^{y} = b^{x+y}

for all `x` and `y`.
This is the single most important identity concerning logs
and exponents. Since `e`^{x} is only a special
case of an exponential function, it is also true that
e^{x} e^{y} = e^{x+y}

*Multiplying the Exponents:*
If `b` is any positive real number then

(b^{x})^{y} = b^{xy}

for all `x` and `y`. Again since `e`^{x}
is a special case of an exponential function, it is also true that
(e^{x})^{y} = e^{xy}

*Converting to roots to exponents:*
The `n`th root of `x` is the same as

x^{1/n}

for all positive `x`. Since square roots are a special case
of `n`th roots, this means that
_
Öx = x^{1/2}

In addition:
__
Öe^{x} = e^{x/2}

*Converting to *`e`^{x}* form:*
If `b` is any positive real number then

b^{x} = e^{x ln(b)}

for all `x`. This includes the case where you have `x`^{x}:
x^{x} = e^{x ln(x)}

or if you have `f(x)`^{x}
f(x)^{x} = e^{x ln(f(x))}

or if you have `x`^{f(x)}:
x^{f(x)} = e^{f(x) ln(x)}

or if you have `f(x)`^{g(x)}
f(x)^{g(x)} = e^{g(x) ln(f(x))}

As an example, suppose you had
`(x`^{2} + 1)^{1/x}. That would be the
same as
e^{(1/x) ln(x2 + 1)}

`e`^{x}* is its own derivative:*
The derivative of `e`^{x} is `e`^{x}. This is the
property that makes `e`^{x} special among all other exponential
functions.

`e`^{x}* is always positive:*
You can put in any `x`, positive or negative, and `e`^{x}
will always be greater than zero. When `x` is positive,
`e`^{x} > 1. When `x` is negative,
`e`^{x} < 1. When
`x = 0` then `e`^{x} = 1.

*The log of the product is the sum of the logs:*
Let `b`, `x`, and `y` all be positive real numbers. Then

log_{b}(xy) = log_{b}(x) + log_{b}(y)

This is the most important property of logs.
Since `ln(x) = log`_{e}(x), it is also true that
log_{e}(xy) = ln(xy) = log_{e}(x) + log_{e}(y) = ln(x) + ln(y)

*The log of the reciprocal is the negative of the log:*
For any positive `b`, `x`, and `y`

log_{b}(1/x) = -log_{b}(x)
log_{b}(y/x) = log_{b}(y) - log_{b}(x)

This includes
ln(1/x) = -ln(x)
ln(y/x) = ln(y) - ln(x)

*Concerning multiplying a log by something else:*
Let `b` and `x` be positive and `k` any real number. Then

k log_{b}(x) = log_{b}(x^{k})

This includes
k ln(x) = ln(x^{k})

It also means that
_
log_{b}(Öx) = (1/2)log_{b}(x)

and
_
ln(Öx) = (1/2)ln(x)

*Converting log bases to natural log*
You can compute any base log using the natural log function (that is `ln`)
alone. If `b` and `x` are both positive then

ln(x)
log_{b}(x) = ~~ ~~
ln(b)

*Every log function is the inverse of some exponential function:*
If `b` is any positive real number, then

b^{logb(x)} = log_{b}(b^{x}) = x

The right-hand part of this equation is true for all `x`. The left-hand
part is true only for positive `x`. The functions, `e`^{x}
and `ln(x)` are also inverses of each other.
e^{ln(x)} = ln(e^{x}) = x

The same rules for `x` apply as above.
*The derivative of the natural log is the reciprocal:*
If `x` is positive,
it is always true that the derivative of `ln(x)` is `1/x`.

To find the derivative of logs of other bases, apply the conversion rule.
So for the derivative of `log`_{b}(x) you end up with

1
~~ ~~
x ln(b)

*The natural log can be expressed as a limit:*
For all positive `x`

x^{h} - 1
ln(x) = lim ~~ ~~
h -> 0 h

*You can only take the log of positive numbers:*
If `x` is negative or zero, you **CAN'T** take the log of `x` --
not the natural log or the log of any base. In addition, the base of a log must also
be positive. As `x` approaches zero from above, `ln(x)` tends to
minus infinity. As `x` goes to positive infinity, so does `ln(x)`.
So `ln(x)` has *no limit* as `x` goes to infinity or as `x`
goes to zero.

*Natural log is positive or negative depending
upon whether *`x`* is greater than or less than 1:*
If `x > 1`, then
`ln(x) > 0`.
If `x < 1`, then
`ln(x) < 0`.
If `x = 1` then
`ln(x) = 0`. Indeed the log to *any*
base of `1` is *always* zero.

## Something you Can't Do with Logs

*There is no formula for the log of a sum:*
Don't go saying that `log(a+b)` is equal to
`log(a) log(b)` because this is **NOT TRUE**.

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email me at
*hahn@netsrq.com*