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 nth root of x is the same as
x^{1/n}for all positive x. Since square roots are a special case of nth roots, this means that
_ √x = x^{1/2}In addition:
__ √e^{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}(√x) = (1/2)log_{b}(x)and
_ ln(√x) = (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
log_{b}(x) = |
ln(x) |
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}) = xThe 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}) = xThe 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
1x ln(b)
The natural log can be expressed as a limit: For all positive x
x^{h} - 1 ln(x) = limh -> 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.
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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