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Home
- Algebra and trigonometry
- 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
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
what is the change in momentum of a body?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
physics, biology and chemistry
this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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