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- College algebra
- Exponential and logarithmic
- Logarithmic functions
Key equations
Definition of the logarithmic function |
For
if and only if
|
Definition of the common logarithm |
For
if and only if
|
Definition of the natural logarithm |
For
if and only if
|
Key concepts
- The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
- Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See
[link] .
- Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See
[link] .
- Logarithmic functions with base
can be evaluated mentally using previous knowledge of powers of
See
[link] and
[link] .
- Common logarithms can be evaluated mentally using previous knowledge of powers of
See
[link]
.
- When common logarithms cannot be evaluated mentally, a calculator can be used. See
[link]
.
- Real-world exponential problems with base
can be rewritten as a common logarithm and then evaluated using a calculator. See
[link]
.
- Natural logarithms can be evaluated using a calculator
[link]
.
Section exercises
Verbal
What is a base
logarithm? Discuss the meaning by interpreting each part of the equivalent equations
and
for
A logarithm is an exponent. Specifically, it is the exponent to which a base
is raised to produce a given value. In the expressions given, the base
has the same value. The exponent,
in the expression
can also be written as the logarithm,
and the value of
is the result of raising
to the power of
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How can the logarithmic equation
be solved for
using the properties of exponents?
Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation
and then properties of exponents can be applied to solve for
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Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base
and how does the notation differ?
The natural logarithm is a special case of the logarithm with base
in that the natural log always has base
Rather than notating the natural logarithm as
the notation used is
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Algebraic
For the following exercises, rewrite each equation in exponential form.
Questions & Answers
how does the planets on our solar system orbit
how many Messier objects are there in space
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
Roaul
I want to be an astronomer. That's my dream
Astrit
Who named the the whole galaxy?
what is the dark matter
Richard
what are the factors upon which the atmosphere is stratified
bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
Vansh
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
Aarya
hey can anyone guide me abt international astronomy olympiad
sahil
how can we learn right and true ?
why the moon is always appear in an elliptical shape
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy....
I have so many questions in mind ....
all of scientists of the word they just give opinion only.
but they never think true or false ...
i respect all of them...
I believes whole universe depending
on true ...থিউরি
Govinda
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
w astronomy and cosmology!
Michele
another theory of universe except big ban
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
I think it couldn't possible on earth
Nagalakshmi
in this time i don't Know
Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*).
So it would just barely creep over the horizon.
Christopher
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Source:
OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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