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Exponential and logarithmic functions

Growth and decay, the number e, log rules, and change of base.

bˣ grows by the same factor for each unit of x; its inverse log_b x asks which exponent produces x. The rules log(ab) = log a + log b and log aᵏ = k log a are the exponent rules read backwards. Picture it: the exponential curve and its log are mirror images across y = x. Think it: e is the base whose growth rate equals its value, which is why calculus prefers it.

Isibonelo esisebenza · y = e^x

Algebra · Functions

Graph and analyse e^(x)

y = e^{x}

Isigaba

  1. e^{x}

    An expression in x. Here is what it does.

  2. \frac{d}{dx} = e^{x}

    Derivative (slope).

Bonisa impendulo
e^{x}

Manje Khetha izinkinga, noma ubhale noma udwebe zakho. Isigaba ngasinye, isithombe, impendulo ifihlakele kuze kube ubuza.

Qhubeka usebenza

I-akhawunti emahhala ifaka iziphawulo kulesifundo ngasinye, irekhodi lokuthi yini oqedile, izinkinga zakho ezixazululiwe endaweni eyodwa, nodokotela owaziyo ongabuza mayelana nalekhasi. I-matematics ivulekile kubo bonke, noma ngabe babhalise noma cha.

Bhala Ngena

Amaphawu asetshenziswa lapha

Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.

Indlela: Exponential and logarithmic functions

  1. An expression in x. Here is what it does.
  2. Derivative (slope).

Imibuzo abantu bebuza

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Okuningi Precalculus

Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsPolynomial division and the remainder theoremParametric equations and polar coordinates