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

Simplify, using the laws of logarithms.

\log 9 + \log 6

Q1a

Simplify, using the laws of logarithms.

\log 48 - \log 6

Q1b

Simplify, using the laws of logarithms.

\log_3 7 + \log_3 3

Q1c

Simplify, using the laws of logarithms. Q1d

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

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

Q3a

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

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

Q3b

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

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

Q3c

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

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

Q3d

Evaluate, using the product law of logarithms.

\displaystyle \log_618 + \log_62 

Q4a

Evaluate, using the product law of logarithms.

\displaystyle \log 40 + \log 2.5 

Q4b

Evaluate, using the product law of logarithms.

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

Q4c

Evaluate, using the product law of logarithms.

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

Q4d

Evaluate, using the product law of logarithms.

\displaystyle \log_354 - \log_32 

Q5a

Evaluate, using the product law of logarithms.

\displaystyle \log 50 000 + \log 5 

Q5b

Evaluate, using the product law of logarithms.

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

Q5c

Evaluate, using the product law of logarithms.

\displaystyle \log 2 + \log 200 

Q5d

Evaluate, using the product law of logarithms.

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

Q6a

Evaluate, using the product law of logarithms.

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

Q6b

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

\displaystyle \log_7(cd) 

Q7a

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

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

Q7b

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

\displaystyle \log(uv^3) 

Q7c

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

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

Q7d

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

\displaystyle \log_210 

Q7e

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

\displaystyle \log_550 

Q7f

Simplify. State any restrictions on the variables.

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

Q9a

Simplify. State any restrictions on the variables.

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

Q9b

Simplify. State any restrictions on the variables.

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

Q9c

Simplify. State any restrictions on the variables.

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

Q9d

Simplify. State any restrictions on the variables.

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

Q10a

Simplify. State any restrictions on the variables.

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

Q10b

Simplify. State any restrictions on the variables.

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

Q10c

Simplify. State any restrictions on the variables.

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

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.
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?
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.

Q12

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

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.

Q14b

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

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?

Q14d

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

Q17

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

\displaystyle \log_4192 -\log_4 3 

Q19a

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

\displaystyle \log_535 -\log_5 7 + \log_525 

Q19b

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

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

Q19c

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

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

Q19d

Prove the quotient law of logarithms by applying algebraic reasoning.

Q20

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

Q21a

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

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?

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?

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.

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?

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