base

(noun)

A number raised to the power of an exponent.

Related Terms

  • Base
  • exponent
  • logarithm
  • root
  • rational number
  • operation

(noun)

In an exponential expression, the value that is multiplied by itself.

Related Terms

  • Base
  • exponent
  • logarithm
  • root
  • rational number
  • operation

Examples of base in the following topics:

  • Changing Logarithmic Bases

    • A logarithm written in one base can be converted to an equal quantity written in a different base.
    • Most common scientific calculators have a key for computing logarithms with base $10$, but do not have keys for other bases.
    • Fortunately, there is a change of base formula that can help.
    • The change-of-base formula can be applied to it:
    • Use the change of base formula to convert logarithms to different bases
  • Logarithmic Functions

    • Logarithms have the following structure: $log{_b}(x)=c$ where $b$ is known as the base, $c$ is the exponent to which the base is raised to afford $x$.
    • The base $b>0$.
    • A logarithm with a base of $10$ is called a common logarithm and is denoted simply as $logx$.
    • A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$.
    • A logarithm with a base of $2$ is called a binary logarithm.
  • Introduction to Exponents

    • Exponential form, written $b^n$, represents multiplying the base $b$ times itself $n$ times.
    • Let's look at an exponential expression with 2 as the base and 3 as the exponent:
    • This means that the base 2 gets multiplied by itself 3 times:
    • Let's look at another exponential expression, this time with 3 as the base and 5 as the exponent:
    • This means that the base 3 gets multiplied by itself 5 times:
  • Simplifying Expressions of the Form log_a a^x and a(log_a x)

    • When the base is the same as the number being modified, the solution is 1, or:
    • In prose, we can say that taking the x-th power of a and then the base-a logarithm gives back x.
    • Conversely, if a positive number a is raised to the power of the log base-a of x, the answer again yields x:
    • Therefore, the logarithm to base-a is the inverse function of
    • Graph of log base 2 .
  • Natural Logarithms

    • The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
    • The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
    • The natural logarithm is the logarithm with base equal to e.
    • Just as the exponential function with base $e$ arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base $e$, also arises in naturally in many contexts.
    • The graph of the natural logarithm lies between the base 2 and the base 3 logarithms.
  • Common Bases of Logarithms

    • Any positive number can be used as the base of a logarithm but certain bases ($10$, $e$, and $2$) have more widespread applications than others.
    • Some bases have more applications than others.
    • Out of the infinite number of possible bases, three stand out as particularly useful.
    • A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$.
    • A logarithm with a base of $2$ is called a binary logarithm and is denoted $ldn$.
  • Converting between Exponential and Logarithmic Equations

    • Here since the bases are both $5$, the exponents are equal.
    • Here the bases are not equal, but it is possible to write 81 using a base of 3 as follows:
    • While you can take the log to any base, it is common to use the common log with a base of $10$ or the natural log with the base of $e$.
    • Here we cannot easily write $17$ with a base of $2$ so instead we take the log of both sides as follows.
    • Here we will use the natural logarithm instead to illustrate the fact that any base will do.
  • Simplifying Exponential Expressions

    • Multiplying exponential expressions with the same base ($a^m \cdot a^n = a^{m+n}$)
    • Applying the rule for dividing exponential expressions with the same base, we have:
    • To simplify the first part of the expression, apply the rule for multiplying two exponential expressions with the same base:
  • Rules for Exponent Arithmetic

    • The following four rules, also known as "identities," hold for all integer exponents, provided that the base is non-zero.
    • Note that you can only add exponents in this way if the corresponding terms have the same base.
    • If you think about an exponent as telling you that you have a certain number of factors of the base, then ${({a}^{n})}^{m}$ means that you have factors $m$ of $a^n$.
    • Notice that two of the terms in this expression have the same base: 2.
    • These two terms can be combined by applying the rule for multiplying exponential expressions with the same base:
  • Graphs of Logarithmic Functions

    • Recall that this is the common log and has a base of $10$.
    • When graphing with a calculator, we use the fact that the calculator can compute only common logarithms (base is $10$), natural logarithms (base is $e$) or binary logarithms (base is $2$).
    • Thus far we have graphed logarithmic functions whose bases are greater than $1$.
    • If we instead consider logarithmic functions with a base $b$, such that $0
    • The graph of the logarithmic function with base $3$ can be generated using the function's inverse.
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