Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication.  Logs "undo" exponentials. Logs are the inverses of exponentials. 

  y = b^x >     is equivalent to     log_b(y) = x 

On the left-hand side above is the exponential statement "y = b^x".  On the right-hand side above, "log_b(y)"  is the equivalent logarithmic statement which is pronounced as "log-base-b of y"; "b" is called "the base of the logarithm", just as b is the base in the exponential expression "b^x".  And, just as the base b in an exponential is always positive and not equal to 1, so also the base b for a logarithm is always positive and not equal to 1.

Whatever is inside the logarithm is called the "argument" of the log.

Note that the base in both the exponential equation and the log equation (above) is "b", but that the x and y switch sides when you switch between the two equations.

Log-base-10 is often used. The base-10 log is usually called "the common log", and is usually written as "log(x)".  That is, if there is no base written, you should assume that the base is 10.

(This is similar to the case for radicals, where, if there is no little number in the front of the radical sign, you know that they mean "square root".  Just as they do not customarily put the little "2" in for the square root, so also they do not customarily put the little "10" in for the common log.)

The other important log is the "natural", or base-e, log, denoted as "ln(x)" and pronounced as "ell-enn-of-x".

Calculate log(100).

Since 100 = 10², then log(100) = log(10²) = 2, because "log(100) = y" means "10 ^y = 100 = 10²", so y = 2.

Calculate ln(e^4.5).

Remember that "ln( )" means the base-e log, so "ln(e^4.5)" might be thought of as "log_e(e^4.5)".  The Relationship says that "ln(e^4.5) = y" means "e ^y = e^4.5", so y = 4.5, and:

ln(e^4.5) = 4.5

Plug "ln(e^4.5)" into your calculator, and you'll get the same answer.  (Make sure you put parentheses around the "e^4.5", so the calculator knows that the exponent is inside the log.)

Calculate ln(2).

Since 2 is a  whole number and since e isn't, then it is unlikely that 2 is a power of e.  So I can't simplify this expression by being clever with The calculator approximate answer is ln(2) = 0.69314718056..., or:

ln(2) = 0.69, rounded to two decimal places.

Logarithm Rules 

1) log_b(mn) = log_b(m) + log_b(n)

2) log_b(^m/_n) = log_b(m) – log_b(n)

3) log_b(mⁿ) = n · log_b(m)

1)  Multiplication inside the log can be turned into addition outside the log, and vice versa.

2)  Division inside the log can be turned into subtraction outside the log, and vice versa.

3)  An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.

(Note:  Just as when you're dealing with exponents, the above rules work only if the bases are the same.  For instance, the expression "log_d(m) + log_b(n)" cannot be simplified, because the bases are not the same, just as x² · y³ cannot be simplified.)

Laws of Exponents

xa xb = xa+b	(xa)b = xab	xa / xb = xa-b
(x y)b =xb yb	x0 = 1	x-a  = 1 / xa