# Describe the quotient rule for logarithms and give an example.

Question
Logarithms
Describe the quotient rule for logarithms and give an example.

2021-02-03
The quotient rule for logarithms with base b is given by: $$\displaystyle{\log{{b}}}{x}-{\log{{b}}}{y}={\log{{b}}}{\left(\frac{{x}}{{y}}\right)}$$
For example, $$\displaystyle{\log{{20}}}-{\log{{5}}}={\log{{\left(\frac{{20}}{{5}}\right)}}}={\log{{4}}}$$

### Relevant Questions

The close connection between logarithm and exponential functions is used often by statisticians as they analyze patterns in data where the numbers range from very small to very large values. For example, the following table shows values that might occur as a bacteria population grows according to the exponential function P(t)=50(2t):
Time t (in hours)012345678 Population P(t)501002004008001,6003,2006,40012,800
a. Complete another row of the table with values log (population) and identify the familiar function pattern illustrated by values in that row.
b. Use your calculator to find log 2 and see how that value relates to the pattern you found in the log P(t) row of the data table.
c. Suppose that you had a different set of experimental data that you suspected was an example of exponential growth or decay, and you produced a similar “third row” with values equal to the logarithms of the population data.
How could you use the pattern in that “third row” to figure out the actual rule for the exponential growth or decay model?
Condense them to the same base before solving for x $$\displaystyle{\log{{16}}}{\left({x}\right)}+{\log{{4}}}{\left({x}\right)}+{\log{{2}}}{\left({x}\right)}={7}$$
Solve for x. $$\displaystyle{\log{{7}}}{\left({x}+{6}\right)}={0}$$
Solve for x $$\displaystyle{1200}{\left({1.03}\right)}^{{x}}={800}{\left({1.08}\right)}^{{x}}$$
Solve for X $$\displaystyle{\log{{X}}}={4}$$
Solve for x using $$\log50=600e^{-0.4x}$$
Solve the equations and inequalities: $$\frac{2^{x}}{3}\leq\frac{5^{x}}{4}$$
Solve the equations and inequalities. Write the solution sets to the inequalities in interval notation. $$\displaystyle{\log{{2}}}{\left({3}{x}−{1}\right)}={\log{{2}}}{\left({x}+{1}\right)}+{3}$$
Solve the equations and inequalities: $$\displaystyle{\frac{{{2}^{{{x}}}}}{{{3}}}}\leq{\frac{{{5}^{{{x}}}}}{{{4}}}}$$