Explicit Functions

If a variable say y, can be expressed explicitly as a function of another variable say x, as y=f(x), then y is known as an explicit function of x, to calculate an output, given input values for the function's rules for the functions arguments. An explicit function can be defined as the dependent variable. It can be written in terms of the independent variable of explicit functions. For eg, The following are explicit functions: x = y2 – 3, f(x) = sqrt (x+7) and x = log2 y.

Implicit Function or Relation of Explicit Equations:

The isolated dependent variables are not mentioned in one side of the expression or equation.

For eg,

The following equation y2 + xy – x2 = 1 represent an implicit relation.

Square Root - Explicit Functions:

A nonnegative number which is multiplied two times itself to give the given number. In general, the square root of y is written `sqrt(y)` or x1/2.

For eg,

Sqrt(9) = 3, since 32 = 9.

Note: sqrt(y) is a non-negative number. If y is a negative value, then the root value of y is imaginary.

Logarithm:

The logarithm base b of a number y is the power to which b must be raised in order to equal y. it can be written as log b y.

For eg,

log2 8 = 3.

Logarithm Rules:

The Algebra properties are followed while doing logarithms.

Assume x, y, a, and b are all positive and a & b are not equal to 1.

Definitions - Functions:

log a x = N, since a*N = x.

log x = log10 x. if a logarithm is written without representing a base it refers general logarithm.

ln x = loge x, since e = 2.718. the log rules are similar to ln.

Note: ln x is equals to Ln x or LN x.

Rules:

Inverse properties:

loga ax = x & a(loga x) = x

Product:

loga (xy) = loga x + loga y.

Quotient:

Log(x/y) = logax – logay.

Power:

loga (xp) = p loga x

Change of base formula:

Log ax = logb x/logb a.

Careful:

Log a (x + y) ≠ log a x + log a y

Log a (x – y) ≠ log a x – log a y

Common Logarithm:

The common logarithm have 10 as its base.

For eg,

log 100 is 2 since 102 = 100.

Natural Logarithm:

The natural logarithm has e as its base. The base e = 2.718

Change of Base Formula - Explicit Functions:

Loga x = logb x/logb a, a & b are not equal to 1.

Examples - Explicit Functions:

Example 1:

Log16 32 = log2 32/log2 16 = 5/4.

Example 2:

Log2 3= log10 3/log10 3

= 0.47712/0.30103

= 1.585

21.585 = 3

Example 3:

Log8 x = ln x/ln 8 = ln x/ln 8.