7.3 Product and Quotient Laws of Logarithms
Chapter
Chapter 7
Section
7.3
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Solutions 52 Videos

Simplify, using the laws of logarithms.

\log 9 + \log 6

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Q1a

Simplify, using the laws of logarithms.

\log 48 - \log 6

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Q1b

Simplify, using the laws of logarithms.

\log_3 7 + \log_3 3

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Q1c

Simplify, using the laws of logarithms.

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Q1d

Simplify each algebraic expression. State any restrictions on the variables.

\log x + \log y + \log (2z)

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Q3a

Simplify each algebraic expression. State any restrictions on the variables.

\log_2a + \log_2(3b) - \log_2(2c)

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Q3b

Simplify each algebraic expression. State any restrictions on the variables.

2\log m + 3\log n - 4\log y

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Q3c

Simplify each algebraic expression. State any restrictions on the variables.

2 \log u + \log v + \frac{1}{2}\log w

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Q3d

Evaluate, using the product law of logarithms.

\displaystyle \log_618 + \log_62

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Q4a

Evaluate, using the product law of logarithms.

\displaystyle \log 40 + \log 2.5

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Q4b

Evaluate, using the product law of logarithms.

\displaystyle \log_{12} 8 + \log_{12}2 + \log_{12}9

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Q4c

Evaluate, using the product law of logarithms.

\displaystyle \log 5 + \log 40 + \log 9

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Q4d

Evaluate, using the product law of logarithms.

\displaystyle \log_354 - \log_32

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Q5a

Evaluate, using the product law of logarithms.

\displaystyle \log 50 000 + \log 5

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Q5b

Evaluate, using the product law of logarithms.

\displaystyle \log_{4} 320 + \log_{4}5

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Q5c

Evaluate, using the product law of logarithms.

\displaystyle \log 2 + \log 200

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Q5d

Evaluate, using the product law of logarithms.

\displaystyle 3\log_{16}2 + 2\log_{16}8 - \log_{16}2

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Q6a

Evaluate, using the product law of logarithms.

\displaystyle \log 20 + \log 2 +\frac{1}{3}\log 125

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Q6b

Write as a sum or difference of logarithms. Simplify, if possible.

\displaystyle \log_7(cd)

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Q7a

Write as a sum or difference of logarithms. Simplify, if possible.

\displaystyle \log_{3}(\frac{m}{n})

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Q7b

Write as a sum or difference of logarithms. Simplify, if possible.

\displaystyle \log(uv^3)

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Q7c

Write as a sum or difference of logarithms. Simplify, if possible.

\displaystyle \log(\frac{a\sqrt{b}}{c^2})

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Q7d

Write as a sum or difference of logarithms. Simplify, if possible.

\displaystyle \log_210

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Q7e

Write as a sum or difference of logarithms. Simplify, if possible.

\displaystyle \log_550

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Q7f

Simplify. State any restrictions on the variables.

\displaystyle \log(\frac{x^2}{\sqrt{x}})

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Q9a

Simplify. State any restrictions on the variables.

\displaystyle \log(\frac{\sqrt{m}}{m^3}) +\log(\sqrt{m})^7

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Q9b

Simplify. State any restrictions on the variables.

\displaystyle \log \sqrt{k} + \log(\sqrt{k})^3 + \log \sqrt[3]{k^2}

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Q9c

Simplify. State any restrictions on the variables.

\displaystyle 2\log w + 3\log\sqrt{w} + \frac{1}{2}\log w^2

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Q9d

Simplify. State any restrictions on the variables.

\displaystyle \log(x^2 -4) - \log(x - 2)

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Q10a

Simplify. State any restrictions on the variables.

\displaystyle \log(x^2 + 7x +12) - \log(x +3)

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Q10b

Simplify. State any restrictions on the variables.

\displaystyle \log(x^2 - x -6) - \log(2x - 6)

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Q10c

Simplify. State any restrictions on the variables.

\displaystyle \log(x^2 + 7x + 12 ) -\log(x^2 - 9)

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Q10d

A certain operational amplifier (Op Amp) produces a voltage output, V_0, in volts, from two input voltage signals, V_1 and V_2, according to the equation V_0 = \log V_2 ? \log V_1.

  • Write a simplified form of this formula, expressing the right side as a single logarithm.
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Q11a

A certain operational amplifier (Op Amp) produces a voltage output, V_0, in volts, from two input voltage signals, V_1 and V_2, according to the equation V_0 = \log V_2 ? \log V_1.

What is the voltage output if

  • i) V_2 is 10 times V_1?
  • ii) V_2 is 100 times V_1?
  • ii) V_2 is equal to V_1?
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Q11b

a) Explain how you can transform the graph of f(x) = \log x to produce g(x) = \log (10nx), for any n > 0.

b) Create two examples to support your explanation. Sketch graphs to illustrate.

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Q12

Graph the functions f(x) = 2 \log x and g(x) = 3 \log x.

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Q14a

Graph the sum of these two functions: p(x) = f(x) + g(x) where f(x) = 2 \log x and g(x) = 3 \log x.

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Q14b

Graph the function q(x) = \log x^5.

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Q14c

How are the functions p(x) = 2 \log x + 3 \log x and q(x) = \log x^5 related? What law of logarithms does this illustrate?

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Q14d

Use the power law of logarithms to verify the product law of logarithms for \log 10^2.

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Q17

Express each as a single logarithm. Then, evaluate, if possible.

\displaystyle \log_4192 -\log_4 3

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Q19a

Express each as a single logarithm. Then, evaluate, if possible.

\displaystyle \log_535 -\log_5 7 + \log_525

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Q19b

Express each as a single logarithm. Then, evaluate, if possible.

\displaystyle \log_a(ab) -\log_a(a^3b)

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Q19c

Express each as a single logarithm. Then, evaluate, if possible.

\displaystyle \log(xy) + \log\frac{y}{x}

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Q19d

Prove the quotient law of logarithms by applying algebraic reasoning.

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Q20

Given the product law of logarithms, prove the product law of exponents.

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Q21a

Given the quotient law of logarithms, prove the quotient law of exponents.

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Q21b

For what positive value of c does the line y = -x + c intersect the circle x^2 + y^2 = 1 at exactly one point?

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Q22

For what value of k does the parabola y = x^2 + k intersect the circle x^2 + y^2 = 1 in exactly three points?

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Q23

A line, l_1, has slope -2 and passes through the point (r, -3). A second line, l_2, is perpendicular to l_1, intersects l_1 at the point (a, b), and passes through the point (6, r). What is the value of a in terms of r.

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Q24

A grocer has c pounds of coffee divided equally among k sacks. She finds n empty sacks and decides to redistribute the coffee equally among the k + n sacks. When this is done, how many fewer pounds of coffee does each of the original sacks hold?

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Q25
Lectures 7 Videos

Introduction to Product Law of Logs

\displaystyle \log_a(AB) =\log_a(A) +\log_a(B)

ex Expand the log and simplify. \displaystyle \log_26

ex Expand the log and simplify. \displaystyle \log_2(12x^2y)

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4.20mins
Introduction to Product Law of Logs

\displaystyle \log_a(\frac{A}{B}) =\log_a(A) -\log_a(B)

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2.08mins
Log Law 2 Quotient Rule