Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function [i]y[/i] = [i]a[/i]x is [i]x[/i] = [i]a[/i]y. The logarithmic function [i]y[/i] = loga[i]x[/i] is defined to be equivalent to the exponential equation [i]x[/i] = [i]a[/i]y. [i]y[/i] = loga[i]x[/i] only under the following conditions: [i]x[/i] = [i]a[/i]y, [i]a[/i] > 0, and [i]a[/i]≠1. It is called the logarithmic function with base [i]a[/i]. Consider what the inverse of the exponential function means: [i]x[/i] = [i]a[/i]y. Given a number [i]x[/i] and a base [i]a[/i], to what power [i]y[/i] must [i]a[/i] be raised to equal [i]x[/i]?[br][br] This unknown exponent, [i]y[/i], equals loga[i]x[/i]. So you see a logarithm is nothing more than an exponent. By definition, [i]a[/i]logax = [i]x[/i], for every real [i]x[/i] > 0.Below are pictured graphs of the form [i]y[/i] = loga[i]x[/i] when [i]a[/i] > 1 and when 0 < [i]a[/i] < 1. Notice that the domain consists only of the positive real numbers and that the function always increases as [i]x[/i] increases.[br][br][br][img]http://img.sparknotes.com/figures/A/a788b9ada7443c462343e2856b0bc3a2/logarithmic.gif[/img][br][br]Figure : Two graphs of [i]y[/i] = loga[i]x[/i]. On the left, [i]y[/i] = log10[i]x[/i], and on the right, [i]y[/i] = log[img]http://img.sparknotes.com/figures/A/a788b9ada7443c462343e2856b0bc3a2/latex_img18.gif[/img][i]x[/i].The domain of a logarithmic function is real numbers greater than zero, and the range is real numbers. The graph of [i]y[/i] = loga[i]x[/i] is symmetrical to the graph of [i]y[/i] = [i]a[/i]x with respect to the line [i]y[/i] = [i]x[/i]. This relationship is true for any function and its inverse.[br]

[size=150]Here are the basic rules for exponents and logarithms:[br][br]**Exponent Rules:**[br][br]1. Product Rule: When you multiply two numbers with the same base, you can add their exponents.[br] a[sup]m[/sup] * a[sup]n[/sup] = a[sup](m+n)[/sup][br][br]2. Quotient Rule: When you divide two numbers with the same base, you can subtract their exponents.[br] a[sup]m[/sup] / a[sup]n[/sup] = a[sup](m-n)[/sup][br][br]3. Power Rule: When you raise a number with an exponent to another exponent, you can multiply the exponents.[br] (a[sup]m[/sup])[sup]n[/sup] = a[sup](m*n)[/sup][br][br]4. Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1.[br] a[sup]0[/sup] = 1 (where 'a' is non-zero)[br][br]5. Negative Exponent Rule: A non-zero number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent.[br] a[sup](-n)[/sup] = 1 / a[sup]n[/sup][br][br]6. Fractional Exponent Rule: A number raised to a fractional exponent is equivalent to taking the root of the number.[br] a[sup](m/n)[/sup] = n√(a[sup]m[/sup])[br][br]**Logarithm Rules:**[br][br]1. Logarithm Definition: The logarithm of a number 'x' to the base 'b' (log_b(x)) is the exponent to which 'b' must be raised to obtain 'x'. In equation form:[br] log[sub]b[/sub](x) = y if and only if b[sup]y[/sup] = x[br][br]2. Logarithm of a Product: The logarithm of a product is equal to the sum of the logarithms of the individual factors.[br] log[sub]b[/sub](x * y) = log[sub]b[/sub](x) + log[sub]b[/sub](y)[br][br]3. Logarithm of a Quotient: The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.[br] log[sub]b[/sub](x / y) = log[sub]b[/sub](x) - log[sub]b[/sub](y)[br][br]4. Logarithm of a Power: The logarithm of a number raised to a power is equal to the exponent times the logarithm of the base.[br] log[sub]b[/sub](x[sup]y[/sup]) = y * log[sub]b[/sub](x)[br][br]5. Change of Base Formula: If you need to find the logarithm of a number with a base different from the one available on your calculator, you can use the change of base formula:[br] - log[sub]b[/sub](x) = log[sub]c[/sub](x) / log[sub]c[/sub](b), where 'c' can be any positive base.[br][br]These rules are fundamental in solving exponential and logarithmic equations and manipulating expressions involving exponents and logarithms.[/size]