6.5 Logarithmic properties (Page 3/10)

College algebra

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Using the quotient rule for logarithms

Expand $\log_{2} (\frac{15 x (x - 1)}{(3 x + 4) (2 - x)}) .$

First we note that the quotient is factored and in lowest terms, so we apply the quotient rule.

\log_{2} (\frac{15 x (x - 1)}{(3 x + 4) (2 - x)}) = \log_{2} (15 x (x - 1)) - \log_{2} ((3 x + 4) (2 - x))

Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that the prime factors of the factor 15 are 3 and 5.

\begin{array}{l} \log_{2} (15 x (x - 1)) - \log_{2} ((3 x + 4) (2 - x)) = [\log_{2} (3) + \log_{2} (5) + \log_{2} (x) + \log_{2} (x - 1)] - [\log_{2} (3 x + 4) + \log_{2} (2 - x)] \\ = \log_{2} (3) + \log_{2} (5) + \log_{2} (x) + \log_{2} (x - 1) - \log_{2} (3 x + 4) - \log_{2} (2 - x) \end{array}

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Expand $\log_{3} (\frac{7 x^{2} + 21 x}{7 x (x - 1) (x - 2)}) .$

$\log_{3} (x + 3) - \log_{3} (x - 1) - \log_{3} (x - 2)$

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Using the power rule for logarithms

We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as $x^{2} ?$ One method is as follows:

\begin{array}{l} \log_{b} (x^{2}) & = \log_{b} (x \cdot x) \\ = \log_{b} x + \log_{b} x \\ = 2 \log_{b} x \end{array}

Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms , which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,

\begin{array}{l} 100 = 10^{2} & \sqrt{3} = 3^{\frac{1}{2}} & \frac{1}{e} = e^{- 1} \end{array}

A General Note

The power rule for logarithms

The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

\log_{b} (M^{n}) = n \log_{b} M

How To

Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.

Express the argument as a power, if needed.
Write the equivalent expression by multiplying the exponent times the logarithm of the base.

Expanding a logarithm with powers

Expand $\log_{2} x^{5} .$

The argument is already written as a power, so we identify the exponent, 5, and the base, $x,$ and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

\log_{2} (x^{5}) = 5 \log_{2} x

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Expand $\ln x^{2} .$

$2 \ln x$

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Rewriting an expression as a power before using the power rule

Expand $\log_{3} (25)$ using the power rule for logs.

Expressing the argument as a power, we get $\log_{3} (25) = \log_{3} (5^{2}) .$

Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

\log_{3} (5^{2}) = 2 \log_{3} (5)

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Expand $\ln (\frac{1}{x^{2}}) .$

$- 2 \ln (x)$

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Using the power rule in reverse

Rewrite $4 \ln (x)$ using the power rule for logs to a single logarithm with a leading coefficient of 1.

Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression $4 \ln (x),$ we identify the factor, 4, as the exponent and the argument, $x,$ as the base, and rewrite the product as a logarithm of a power: $4 \ln (x) = \ln (x^{4}) .$

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Rewrite $2 \log_{3} 4$ using the power rule for logs to a single logarithm with a leading coefficient of 1.

$\log_{3} 16$

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Expanding logarithmic expressions

Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example:

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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