For any algebraic expressions
and
and any positive real number
where
if and only if
Definition of a logarithm
For any algebraic expression
S and positive real numbers
and
where
if and only if
One-to-one property for logarithmic functions
For any algebraic expressions
S and
T and any positive real number
where
if and only if
Key concepts
We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See
[link] .
When we are given an exponential equation where the bases are
not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See
[link] ,
[link] , and
[link] .
When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See
[link] .
We can solve exponential equations with base
by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See
[link] and
[link] .
After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See
[link] .
When given an equation of the form
where
is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation
and solve for the unknown. See
[link] and
[link] .
We can also use graphing to solve equations with the form
We graph both equations
and
on the same coordinate plane and identify the solution as the
x- value of the intersecting point. See
[link] .
When given an equation of the form
where
and
are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation
for the unknown. See
[link] .
Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See
[link] .
Section exercises
Verbal
How can an exponential equation be solved?
Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.
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
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
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
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
hello
Jackson
hi
Elyana
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
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
by big bang
universal
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
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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