6.3 Logarithmic functions (Page 2/9)

College algebra

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We can express the relationship between logarithmic form and its corresponding exponential form as follows:

\log_{b} (x) = y \Leftrightarrow b^{y} = x, b > 0, b \neq 1

Note that the base $b$ is always positive.

Because logarithm is a function, it is most correctly written as $\log_{b} (x),$ using parentheses to denote function evaluation, just as we would with $f (x) .$ However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as $\log_{b} x .$ Note that many calculators require parentheses around the $x .$

We can illustrate the notation of logarithms as follows:

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means $y = \log_{b} (x)$ and $y = b^{x}$ are inverse functions.

A General Note

Definition of the logarithmic function

A logarithm base $b$ of a positive number $x$ satisfies the following definition.

For $x > 0, b > 0, b \neq 1,$

y = \log_{b} (x) is equivalent to b^{y} = x

where,

we read $\log_{b} (x)$ as, “the logarithm with base $b$ of $x$ ” or the “log base $b$ of $x . "$
the logarithm $y$ is the exponent to which $b$ must be raised to get $x .$

Also, since the logarithmic and exponential functions switch the $x$ and $y$ values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

the domain of the logarithm function with base $b is (0, \infty) .$
the range of the logarithm function with base $b is (- \infty, \infty) .$

Q&A

Can we take the logarithm of a negative number?

No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.

How To

Given an equation in logarithmic form $\log_{b} (x) = y,$ convert it to exponential form.

Examine the equation $y = \log_{b} x$ and identify $b, y, and x .$
Rewrite $\log_{b} x = y$ as $b^{y} = x .$

Converting from logarithmic form to exponential form

Write the following logarithmic equations in exponential form.

$\log_{6} (\sqrt{6}) = \frac{1}{2}$
$\log_{3} (9) = 2$

First, identify the values of $b, y, and x .$ Then, write the equation in the form $b^{y} = x .$

$\log_{6} (\sqrt{6}) = \frac{1}{2}$
Here, $b = 6, y = \frac{1}{2}, and x = \sqrt{6.}$ Therefore, the equation $\log_{6} (\sqrt{6}) = \frac{1}{2}$ is equivalent to $6^{\frac{1}{2}} = \sqrt{6} .$
$\log_{3} (9) = 2$
Here, $b = 3, y = 2, and x = 9.$ Therefore, the equation $\log_{3} (9) = 2$ is equivalent to $3^{2} = 9.$

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Write the following logarithmic equations in exponential form.

$\log_{10} (1, 000, 000) = 6$
$\log_{5} (25) = 2$

$\log_{10} (1, 000, 000) = 6$ is equivalent to $10^{6} = 1, 000, 000$
$\log_{5} (25) = 2$ is equivalent to $5^{2} = 25$

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Converting from exponential to logarithmic form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base $b,$ exponent $x,$ and output $y .$ Then we write $x = \log_{b} (y) .$

Converting from exponential form to logarithmic form

Write the following exponential equations in logarithmic form.

$2^{3} = 8$
$5^{2} = 25$
$10^{- 4} = \frac{1}{10,000}$

First, identify the values of $b, y, and x .$ Then, write the equation in the form $x = \log_{b} (y) .$

$2^{3} = 8$
Here, $b = 2,$ $x = 3,$ and $y = 8.$ Therefore, the equation $2^{3} = 8$ is equivalent to $\log_{2} (8) = 3.$
$5^{2} = 25$
Here, $b = 5,$ $x = 2,$ and $y = 25.$ Therefore, the equation $5^{2} = 25$ is equivalent to $\log_{5} (25) = 2.$
$10^{- 4} = \frac{1}{10,000}$
Here, $b = 10,$ $x = - 4,$ and $y = \frac{1}{10,000} .$ Therefore, the equation $10^{- 4} = \frac{1}{10,000}$ is equivalent to ${log}_{10} (\frac{1}{10,000}) = - 4.$

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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.

Kate Reply

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?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

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

Maria Reply

please solve

Sharon

8m/s²

Aishat

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

Saheed Reply

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

Ramon Reply

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

Scratch that

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

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

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

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

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?

Nassze Reply

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.

JALLAH Reply

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

Destiny Reply

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, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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