exponent

(noun)

the power to which a number, symbol or expression is to be raised: for example, the $3$ in $x^3$.

Related Terms

Examples of exponent in the following topics:

  • Expressing Functions as Power Functions

    • Since all infinitely differentiable functions can be represented in power series, any infinitely differentiable function can be represented as a sum of many power functions (of integer exponents).
    • Therefore, an arbitrary function that is infinitely differentiable is expressed as an infinite sum of power functions ($x^n$) of integer exponent.
    • As more power functions with larger exponents are added, the Taylor polynomial approaches the correct function.

  • Curve Sketching

    • If the exponent of $x$ is always even in the equation of the curve, then the $y$-axis is an axis of symmetry for the curve.
    • Similarly, if the exponent of $y$ is always even in the equation of the curve, then the $x$-axis is an axis of symmetry for the curve.

  • Derivatives of Logarithmic Functions

    • Applying the chain rule and the property of exponents we derived earlier, we can take the derivative of both sides:
    • We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such as products with many terms, quotients of composed functions, or functions with variable or function exponents.

  • Logarithmic Functions

    • The logarithm of a number is the exponent by which another fixed value must be raised to produce that number.
    • The logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number.
    • A naive way of defining the logarithm of a number x with respect to base b is the exponent by which b must be raised to yield x.

  • Solving Differential Equations

    • The "degree" of a differential equation, similarly, is determined by the highest exponent on any variables involved.

  • Linear and Quadratic Functions

    • The expression $ax^2+bx+c$ in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of $x$ is 2.

  • Separable Equations

    • ., combine all possible terms, rewrite any logarithmic terms in exponent form, and express any arbitrary constants in the most simple terms possible).

  • Exponential and Logarithmic Functions

    • The logarithm of a number $x$ with respect to base $b$ is the exponent by which $b$ must be raised to yield $x$.