8.4 Laws of Logarithms
Chapter
Chapter 8
Section
8.4
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) 

4.20mins
Introduction to Product Law of Logs

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

2.08mins
Log Law 2 Quotient Rule
Solutions 59 Videos

Write the expression as a sum of difference of logarithms.

 \displaystyle \log(45 \times 68) 

1.07mins
Q1a

Write the expression as a sum of difference of logarithms.

 \displaystyle \log_mpq 

0.15mins
Q1b

Write the expression as a sum of difference of logarithms.

 \displaystyle \log (\frac{123}{31}) 

0.29mins
Q1c

Write the expression as a sum of difference of logarithms.

 \displaystyle \log_m(\frac{p}{q}) 

0.14mins
Q1d

Write the expression as a sum of difference of logarithms.

 \displaystyle \log_{2}(14 \times 9) 

0.37mins
Q1e

Write the expression as a difference of logarithms.

 \displaystyle \log_4 (\frac{81}{30}) 

1.15mins
Q1f

Express each of the following as a logarithm of a product of quotient.

\displaystyle \log 5 + \log 7 

0.15mins
Q2a

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log x - \log y 

0.14mins
Q2b

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_34 - \log_32 

0.12mins
Q2c

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_67 + \log_68 + \log_69 

0.11mins
Q2d

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_67 + \log_68 + \log_69 

0.32mins
Q2e

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_410 + \log_412 - \log_420 

0.28mins
Q2f

Express each of the following in the form r \log_ax.

 \displaystyle \log 5^2 

0.10mins
Q3a

Express each of the following in the form r \log_ax.

 \displaystyle \log_37^{-1} 

0.07mins
Q3ab

Express each of the following in the form r \log_ax.

 \displaystyle \log_mp^q 

0.14mins
Q3c

Express each of the following in the form r \log_ax.

 \displaystyle \log\sqrt[3]{45} 

0.22mins
Q3d

Express each of the following in the form r \log_ax.

 \displaystyle \log_7(36)^{0.5} 

0.38mins
Q3e

Express each of the following in the form r \log_ax.

 \displaystyle \log_5\sqrt{5}[125] 

0.28mins
Q3f

Use the laws of logarithms to simplify and then evaluate each expression.

 \displaystyle \log_3135 - \log_35 

0.27mins
Q4a

Use the laws of logarithms to simplify and then evaluate each expression.

 \displaystyle \log_510 + \log_52.5 

0.22mins
Q4b

Use the laws of logarithms to simplify and then evaluate each expression.

 \displaystyle \log 50 + \log 2 

0.21mins
Q4c

Use the laws of logarithms to simplify and then evaluate each expression.

 \displaystyle \log_44^7 

0.10mins
Q4d

Use the laws of logarithms to simplify and then evaluate each expression.

 \displaystyle \log_2224 - \log_27 

0.28mins
Q4e

Use the laws of logarithms to simplify and then evaluate each expression.

 \displaystyle \log \sqrt{10} 

0.14mins
Q4f

Write y = \log_2(4x), y = \log_2(8x), and y= \log_2(\frac{x}{2}) with respect to y =\log_2x.

1.28mins
Q5

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_{25}5^3 

0.33mins
Q6a

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_6 54 +\log_62 - \log_63 

0.33mins
Q6b

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_66\sqrt{6} 

0.27mins
Q6c

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_2 \sqrt{36} - \log_2\sqrt{72} 

1.20mins
Q6d

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_354 + \log_3(\frac{3}{2}) 

0.35mins
Q6e

Express each of the following as a logarithm of a product of quotient.

 \displaystyle \log_82 + 3\log_82 + \frac{1}{2} \log_816 

1.00mins
Q6f

Use the laws of logarithms to express each of the following int terms of \log_b x , \log_by, and \log_bz.

 \displaystyle \log_bxyz 

0.27mins
Q7a

Use the laws of logarithms to express each of the following int terms of \log_b x , \log_by, and \log_bz.

 \displaystyle \log_b(\frac{z}{xy}) 

0.39mins
Q7b

Use the laws of logarithms to express each of the following int terms of \log_b x , \log_by, and \log_bz.

 \displaystyle \log_bx^2y^3 

0.35mins
Q7c

Use the laws of logarithms to express each of the following int terms of \log_b x , \log_by, and \log_bz.

 \displaystyle \log_b\sqrt{x^5yz^3} 

1.39mins
Q7d

Explain why \log_53 + \log_5\frac{1}{3} = 0.

0.29mins
Q8

Write each expression as a single log.

 \displaystyle 3\log_{5}2 + \log_{5}7 

0.24mins
Q9a

Write each expression as a single log.

 \displaystyle 2\log_38 - 5\log_32 

0.44mins
Q9b

Write each expression as a single log.

 \displaystyle 2\log_23 + \log_25 

0.31mins
Q9c

Write each expression as a single log.

 \displaystyle \log_3 12 + \log_3 2 -\log_36 

0.40mins
Q9d

Write each expression as a single log.

 \displaystyle \log_43 + \frac{1}{2}\log_48 -\log_42 

1.13mins
Q9e

Write each expression as a single log.

 \displaystyle 2\log 8 + \log 9 - \log 36 

0.44mins
Q9f

Use the laws of logarithms to express each side of the equation as a single logarithm. Then compare both sides of the equation to solve.

 \displaystyle \log_2 x = 2\log_2 7 +\log_2 5 

0.39mins
Q10a

Use the laws of logarithms to express each side of the equation as a single logarithm. Then compare both sides of the equation to solve.

 \displaystyle \log x = 2\log 4 + 3 \log 3 

0.31mins
Q10b

Use the laws of logarithms to express each side of the equation as a single logarithm. Then compare both sides of the equation to solve.

 \displaystyle \log_4x + \log_412 = \log_448 

0.23mins
Q10c

Use the laws of logarithms to express each side of the equation as a single logarithm. Then compare both sides of the equation to solve.

 \displaystyle \log_7x = 2 \log_7 25 - 3\log_7 5 

0.44mins
Q10d

Use the laws of logarithms to express each side of the equation as a single logarithm. Then compare both sides of the equation to solve.

 \displaystyle \log_3x = 2 \log_3 10 - \log_325 

0.28mins
Q10e

Use the laws of logarithms to express each side of the equation as a single logarithm. Then compare both sides of the equation to solve.

 \displaystyle \log_5x - \log_58 = \log_56 + 3\log_52 

0.28mins
Q10f

Write each expression as a single log. Assume that all the variables represent positive numbers

 \displaystyle \log_2 x +\log_2 y + \log_2 z 

0.13mins
Q11a

Write each expression as a single log. Assume that all the variables represent positive numbers

 \displaystyle \log_5 u - \log_5 v +\log_5 w 

0.19mins
Q11b

Write each expression as a single log. Assume that all the variables represent positive numbers

 \displaystyle \log_6a - (\log_6 b + \log_6 c) 

0.26mins
Q11c

Write each expression as a single log. Assume that all the variables represent positive numbers

 \displaystyle \log_2x^2 - \log_2xy + \log_2y^2 

0.38mins
Q11d

Write each expression as a single log. Assume that all the variables represent positive numbers

 \displaystyle 1 +\log_3x^2 

0.20mins
Q11e

Write each expression as a single log. Assume that all the variables represent positive numbers

 \displaystyle 3 \log_4 x + 2 \log_4x - \log_4 y 

0.45mins
Q11f

Write \frac{1}{2} \log_ax + \frac{1}{2}\log_ay - \frac{3}{4} \log_az as a single logarithm. Assume that all the variables represent positive numbers.

0.40mins
Q12

Write g(x) in terms of f(x) when f(x) = \log_2x to the graph g(x) = \log_2(8x^3).

1.02mins
Q13

Explain how the laws of logarithms can help you evaluate \log_3 (\frac{\sqrt[5]{27}}{2187}).

1.10mins
Q15

Explain why \log_xx^{m - 1} + 1 = m.

Draw the graphs of y = \log x + \log 2x and y =\log 2x^2. Although the graph are different, simplifying the first expression using the laws of logarithms produces the second expression. Explain why the graph are different.