base

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.