2.6.1: Preparation M.6
- Page ID
- 148564
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)(1) Fill in the blanks:
(a) If you wanted to solve the equation x + 7 = -2 you could ___________________ from both sides of the equation.
(b) This works because the operation of __________________ “undoes” the operation of
_____________________.
(c) After performing this step, you arrive at the solution, x = _______________.
(2) Fill in the blanks:
(a) If you wanted to solve the equation 3x = 15 you could ___________________ both sides of the
equation by _____________.
(b) This works because the operation of __________________ “undoes” the operation of
_____________________.
(c) After performing this step, you arrive at the solution, x = _______________.
(3) Fill in the blanks:
(a) If you wanted to solve the equation x2 = 15 you could ___________________ both sides of the equation.
(b) This works because the operation of __________________ “undoes” the operation of
_____________________.
(c) After performing this step, you arrive at a solution, x = __________or__________. Round to the nearest hundredth.
Logarithms (base 10)
Now we would like to introduce a new operation which “undoes” exponentiation. That operation is written as log(x), and we say this as “the log of x.”
The term log is an abbreviation for the mathematical term logarithm. A logarithm is a quantity representing the power to which a fixed number (the base, in this case 10) must be raised to produce a given number.
For example, a logarithm, or log operation, can be used to find the exponent to which one must raise the number 10 to get the number x. The expression “log(1000)” is then equal to 3, since 103=1000. Another example is that “log(10)” is equal to 1, since 101=10. (Note: Most calculators [or spreadsheet applications] have a key [or function] labeled “Log” which performs this operation. Note that here, the parentheses do not indicate multiplication; they indicate a process is being done.)
(4) What are the following logs?
(a) log(100) = __________
(b) log(100,000) = __________
Generalizing, since log(1000) = 3 and 103 = 1000 are both true, we can replace the 1000 with 103 in the first equation. This gives us the equation: log(103)= 3. Similarly, log(102)= 2 and log(101)= 1. This leads to the general statement that:
\[ log(10^x) = x \nonumber\]
Now we would like to use the log operation to solve equations. Consider the equation 10x=16. Notice that the exponent is not a whole number in this case. We can still solve this equation by doing something to both sides of the equation, as we did in Questions 1–3. In this case, we will “take the log” of both sides of the equation:
| Initial equation | 10x = 16 |
| Apply logs to both sides of the equation |
log(10x) = log 16 x = log 16 |
| The left side is explained above. The right side is rounded to the thousandth place after using the “Log” key on the calculator to find log 16. |
Answer: x = 1.204 |
We can check this result using a calculator. If x = 1.204, we get 10x = 101.204 = 15.996, which is about 16. (It is not exactly 16 because we rounded log 16 to three decimal places.)
(5) (a) Using your calculator (or spreadsheet application) follow the steps above to solve the equation 10x = 25. Round to the nearest thousandth.
(b) Using the value you found for x in Question 5(a), find 10x. Round to the nearest whole number. Does 10x = 25?
After Preparation M.6 (survey)
You should be able to do the following things for the next collaboration. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Collaboration M.6, you should understand the concepts and demonstrate the skills listed below:
| Skill or Concept: I can … | Rating from 1 to 5 |
| solve linear equations by undoing operations. | |
| understand that the log operation undoes the “raise 10 to the power of…” operation. |