# Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. log5 x

Question
Analyzing functions
Rewrite the logarithm as a ratio of
(a) common logarithms and
(b) natural logarithms. $$\displaystyle{\log{{5}}}{x}$$

2020-11-21
To evaluate logarithms to any base, we use the change-of-base formula: $$\displaystyle{\log{{c}}}{a}=\frac{{{\log{{b}}}{a}}}{{{\log{{b}}}{c}}}$$
a) $$\displaystyle{\log{{5}}}{x}=\frac{{\log{{x}}}}{{\log{{5}}}}$$
This is a common logarithm as the base is changed from 5 to 10 the same way the logarithm is changed from base “c” to base “b” in the formula.
b) $$\displaystyle{\log{{5}}}{x}=\frac{{\ln{{x}}}}{{\ln{{5}}}}$$
To change $$\displaystyle{\log{{5}}}$$ x to ratio of a natural logarithm, change $$\displaystyle{\log{{5}}}$$, also written as ln.

### Relevant Questions

Use the properties of logarithms to rewrite each expression as the logarithm of a single expression. Be sure to use positive exponents and avoid radicals.
a. $$2\ln4x^{3}\ +\ 3\ln\ y\ -\ \frac{1}{3}\ln\ z^{6}$$
b. $$\ln(x^{2}\ -\ 16)\ -\ \ln(x\ +\ 4)$$
Write the function $$\displaystyle{y}=-{\sin{{x}}}$$ as a phase shift of $$\displaystyle{y}={\sin{{x}}}$$.
Find the vertex, focus, and directrix for the parabolas:
a) $$\displaystyle{\left({y}–{9}\right)}^{{2}}={8}{\left({x}-{2}\right)}$$
b) $$\displaystyle{y}^{{2}}–{4}{y}={4}{x}–{2}^{{2}}$$
c) $$\displaystyle{\left({x}–{6}\right)}^{{2}}={4}{\left({y}–{2}\right)}$$
Sketch the graph of the function $$\displaystyle{f{{\left({x}\right)}}}=-{x}^{{3}}+{3}{x}^{{2}}–{7}$$. List the coordinated of where extrema or points of inflection occur. State where the function is increasing or decreasing as well as where it is concave up or concave down.
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?
Find the area of the region below $$\displaystyle{y}={x}^{{2}}–{3}{x}+{4}$$ and above $$\displaystyle{y}={6}{f}{\quad\text{or}\quad}{3}≤{x}≤{6}$$
Find the region enclosed by the curves $$\displaystyle{x}={3}{y}^{{2}}{\quad\text{and}\quad}{x}={y}^{{2}}+{7}$$
List the points of intersection of these curves from bottom to top in the form (x, y).
Find the $$\displaystyle{\left({f}+{g}\right)}{\left({x}\right)}{\quad\text{and}\quad}{\left({f}–{g}\right)}{\left({x}\right)}$$, if $$\displaystyle{f{{\left({x}\right)}}}={10}{x}–{8}{\quad\text{and}\quad}{g{{\left({x}\right)}}}={5}{x}+{7}$$
Graph the equation and identify the y-intercept.
$$\displaystyle{y}={x}+{5}$$
Find the area between the curves $$\displaystyle{y}={e}^{{-{{0.7}}}}{x}{\quad\text{and}\quad}{y}={2.1}{x}+{1}$$ from x = 0 to x =1.