Flex Day 1

During Class

Brain Gains

Match the expanded log expressions in the left column with the condensed log expressions in the right column.

Expanded Expressions | Condensed Expressions |
— | — |
1. \(\log 3 + \log 2\) | A. \(\log 4\) |
2. \(\log 10 - \log 2\) | B. \(\log 5\) |
3. \(3\log 2\) | C. \(\log 6\) |
4. \(\frac{2}{3}\log 27\) | D. \(\log 8\) |
5. \(\log 24 - \log 6\) | E. \(\log 9\) |
6. \(\log 6 + \log 3\) | F. \(\log 18\) | |
Solution
  1. C
  2. B
  3. D
  4. E
  5. A
  6. F

We can use R to calculate approximations for each expression.

log(3,10) + log(2,10)
log(6,10)

log10(10) - log10(2)
log(5,10)

3*log(2,10)
log10(8)

(2/3)*log(27,10)
log(9,10)

log(24,10)-log10(6)
log10(4)

log(6,10)+log10(3)
log(18,10)

Definitions and Reminders

Definition: Exponential Function (OpenStax Textbook)

An exponential function is of the form \(b^x\) where \(b>0\) and \(b \neq 1\).

Note: An exponential function is not an algebraic function.

Key Characteristics of Exponential Function
  • Domain is all real numbers
  • Range is positive real numbers
  • When \(b > 1\) the function is increasing on its entire domain.
    • As \(x\) decreases (approaches negative infinity) the output gets close to zero.
    • As \(x\) increases the output increases without bound.
    • These functions are concave up. (We will learn the precise meaning of concavity later this semester.)
  • When \(0< b < 1\) the function is decreasing on its entire domain.
    • As \(x\) decreases (approaches negative infinity) the output grows without bound.
    • As \(x\) increases the output gets close to zero.
    • These functions are concave up. (We will learn the precise meaning of concavity later this semester.)
Properties of Exponents

If \(a > 0\), \(b >0\), and \(m\) and \(n\) are any real number, then - \(a^m \cdot a^n = a^{m+n}\) - \(\frac{a^m}{a^n} = a^{m-n}\) - \((a^m)^n = a^{mn}\) - \(\frac{1}{a^n} = a^{-n}\) - \(\sqrt[n]{a} = a^{1/n}\) - \(\sqrt[n]{a^m} = a^{m/n}\) - \(a^m \cdot b^m = (a \cdot b)^m\) - \(\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m\)

Definition: Logarithm (OpenStax Textbook)

A logarithm base \(b\) of a positive number satisfies the following definition.

For \(x>0\), \(b>0\), and \(b \neq 1\), \(y = \log_b(x)\) is equivalent to \(b^y = x\)

  • We read \(\log_b(x)\) as, “the log base \(b\) of \(x\)” or “the logarithm with base \(b\) of \(x\)”.
  • The logarithm is the exponent that \(b\) must be raised to get \(x\).
  • When we evaluate \(\log_b(x)\) we answer the question, “To what exponent must \(b\) be raised to in order to get \(x\)?”

The key idea is that we can rewrite any logarithm in exponential form, as \[\huge{y = \log_b(x) \Leftrightarrow b^y = x}\]

Definition: Logarithmic Function

A logarithmic function is of the form \(\log_b(x)\) where \(b>0\) and \(b \neq 1\).

Note: A logarithmic function is not an algebraic function.

Key Characteristics of Logarithmic Function
  • Domain is positive real numbers
  • Range is all real numbers
  • When \(b > 1\) the function is increasing on its entire domain.
    • As \(x\) gets close to zero from the right the output approaches negative infinity.
    • As \(x\) increases the output increases without bound.
    • These functions are concave down.
  • When \(0< b < 1\) the function is decreasing on its entire domain.
    • As \(x\) gets close to zero from the right the output grows without bound (approaches infinity).
    • As \(x\) increases the output approaches negative infinity.
    • These functions are concave up.
Properties of Logarithms

If \(a>0\), \(b > 0\), \(c > 0\), \(b \neq 1\), and \(n\) is any real number, then - \(\log_b b^x = x\) and \(b^{\log_b x} = x\) (inverse property) - Special Cases - \(\log_b b = \log_b b^1 = 1\) - \(\log_b 1 = \log_b b^0 = 0\) - \(\log_b(ac) = \log_b(a) + \log_b(c)\) (product property) - \(\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\) (quotient property) - \(\log_b(a^n) = n\log_b(a)\) (power property)

\(\huge{\text{Logs turn multiplication into addtion!}}\)

Group Meeting

General Instructions - Give each group member a several minutes to share.

Activity - Logs with \(\prod\) and \(\sum\) notation

Write in expanded form. - \(\prod_{m=1}^2(mx + 1)\) - \(\prod_{m=1}^3(mx + 1)\) - \(\sum_{m=1}^3(mx + 1)\) - \(\sum_{m=1}^3\ln(mx + 1)\) - \(\sum_{m=1}^3(x_m + 1)\) - \(\sum_{m=1}^3\ln(x_m + 1)\)

Rewrite the following expressions using log properties. - \(\ln \left(\prod_{m=1}^2(mx + 1)\right)\) - \(\ln \left(\prod_{m=1}^7(mx + 1)\right)\) - \(\ln \left(\prod_{i=1}^5((y_i - 3)^2) \right)\) - \(\ln \left(\prod_{i=1}^n((ax_i+b-y_i)^2) \right)\)

\(\huge{\text{Logs turn multiplication into addtion!}}\)

Activity - Practice with Logarithms

Use R to plot the functions \(f(x) = 2^x\), \(g(x) = e^x\), and \(h(x) = 10^x\). Then answer the following questions. - What is the domain of an exponential function? - What is the range of an exponential function? - Give an example of a power function. What is the difference between a power function and an exponential function?

Use R to plot the function \(F(x) = \log_2(x)\), \(G(x) = \ln(x)\), and \(H(x) = \log(x)\). Then answer the following questions. - What is the domain of a logarithmic function? - What is the range of a logarithmic function?

Activity - Solving Equations

Solve the following equations for \(x\). Write down the exact answer and decimal approximation (round to 4 decimal places). - \(\ln(4x-2) = 5\) - \(\log(5x-30) = 1\) - \(9^x = 49\) - \(2^x = 56\) - \(-2\log_5(7x) = 2\) - \(\log_3(x-3) - 4 = 12\) - \(\log_2(x-1) + \log_2(x+5) = 4\) - \(e^{x-1}-5 = 5\) - \(3(4^{2-5x})-7=8\) - \(\ln(x+5) + \ln(x) = \ln(12x)\)

Between Class Sessions

Skill Practice (KA Homework)

  • Finish any of the first three log assignments if you have not already finished them.
  • Continue working on 1 – Solve Logarithmic Equations (and IVT) assignment

Applied Practice (Project Work)

  • Continue working on Project 1 Task 3