6.5 Logarithmic properties (Page 4/10)

College algebra

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\begin{array}{l} \log_{b} (\frac{6 x}{y}) & = \log_{b} (6 x) - \log_{b} y \\ = \log_{b} 6 + \log_{b} x - \log_{b} y \end{array}

We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:

\begin{array}{l} \log_{b} (\frac{A}{C}) & = \log_{b} (A C^{- 1}) \\ = \log_{b} (A) + \log_{b} (C^{- 1}) \\ = \log_{b} A + (- 1) \log_{b} C \\ = \log_{b} A - \log_{b} C \end{array}

We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.

With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.

Expanding logarithms using product, quotient, and power rules

Rewrite $\ln (\frac{x^{4} y}{7})$ as a sum or difference of logs.

First, because we have a quotient of two expressions, we can use the quotient rule:

\ln (\frac{x^{4} y}{7}) = \ln (x^{4} y) - \ln (7)

Then seeing the product in the first term, we use the product rule:

\ln (x^{4} y) - \ln (7) = \ln (x^{4}) + \ln (y) - \ln (7)

Finally, we use the power rule on the first term:

\ln (x^{4}) + \ln (y) - \ln (7) = 4 \ln (x) + \ln (y) - \ln (7)

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Expand $\log (\frac{x^{2} y^{3}}{z^{4}}) .$

$2 \log x + 3 \log y - 4 \log z$

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Using the power rule for logarithms to simplify the logarithm of a radical expression

Expand $\log (\sqrt{x}) .$

\begin{array}{l} \log (\sqrt{x}) & = \log x^{(\frac{1}{2})} \\ = \frac{1}{2} \log x \end{array}

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Expand $\ln (\sqrt[3]{x^{2}}) .$

$\frac{2}{3} \ln x$

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Q&A

Can we expand $\ln (x^{2} + y^{2}) ?$

No. There is no way to expand the logarithm of a sum or difference inside the argument of the logarithm.

Expanding complex logarithmic expressions

Expand $\log_{6} (\frac{64 x^{3} (4 x + 1)}{(2 x - 1)}) .$

We can expand by applying the Product and Quotient Rules.

\begin{array}{l} \log_{6} (\frac{64 x^{3} (4 x + 1)}{(2 x - 1)}) & = \log_{6} 64 + \log_{6} x^{3} + \log_{6} (4 x + 1) - \log_{6} (2 x - 1) & Apply the Quotient Rule . \\ = \log_{6} 2^{6} + \log_{6} x^{3} + \log_{6} (4 x + 1) - \log_{6} (2 x - 1) & {Simplify by writing  64 as 2}^{6} . \\ = 6 \log_{6} 2 + 3 \log_{6} x + \log_{6} (4 x + 1) - \log_{6} (2 x - 1) & Apply the Power Rule . \end{array}

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Expand $\ln (\frac{\sqrt{(x - 1) {(2 x + 1)}^{2}}}{(x^{2} - 9)}) .$

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

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Condensing logarithmic expressions

We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.

How To

Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm.

Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.
Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.
Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.

Using the product and quotient rules to combine logarithms

Write $\log_{3} (5) + \log_{3} (8) - \log_{3} (2)$ as a single logarithm.

Using the product and quotient rules

\log_{3} (5) + \log_{3} (8) = \log_{3} (5 \cdot 8) = \log_{3} (40)

This reduces our original expression to

\log_{3} (40) - \log_{3} (2)

Then, using the quotient rule

\log_{3} (40) - \log_{3} (2) = \log_{3} (\frac{40}{2}) = \log_{3} (20)

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Condense $\log 3 - \log 4 + \log 5 - \log 6.$

$\log (\frac{3 \cdot 5}{4 \cdot 6});$ can also be written $\log (\frac{5}{8})$ by reducing the fraction to lowest terms.

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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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