Math

• Notes
• HW 2.6
• HW 2.7
• HW 3.1
• HW 3.2
• HW 3.3
• HW 3.4
• HW 3.5
• HW 4.1
• HW 4.2
• HW 4.3
• HW 4.4
• HW 4.5
• Exam November 9th
• Exam November 9th Corrections
• Test Review 03-08-2016

• Chem 7.31 - 7.45

Negative Power Rule

${x}^{-a}=\frac{1}{{x}^{a}}$

Example:

$3^{-2} = \frac{1}{3^2}$

$= \frac{1}{9}$

Asymptotes

Let $n$ be the degree of the numerator, and $m$ the degree of the denominator.

If $n<m$, the horizontal asymptote is the $x$-axis (or $y=0$)

If $n=m$, $y=\frac{a_n}{b_m}$ is the horizontal asymptote.

If $n>m$, the graph has no horiztonal asymptote.

Graph Symmetry

If $f(-x) = f(x)$, the graph has $y$-axis symmetry.
If $f(-x) = -f(x)$, the graph has origin symmetry.

Axis Reflection

$x$-axis Reflection: $y=-b^x$
$y$-axis Reflection: $y=b^{-x}$

Exponential Equations

If $a^x = a^y$, then $x=y$

Logs and Exponents

$\log_bM=\log_bN$
$M=N$

If $y=b^x$, then $x=\log_by$

$\log_bb^n = n$

Quotient Rule
$\log_b(\frac{M}{N}) = \log_bM-\log_bN$

(If $b$, $M$, and $N$ are positive real numbers and $b\ne1$, then)

Product Rule
$\log_b(MN) = \log_bM+\log_bN$

Power Rule
$\log_bM^p = p \log_bM$

Natural Logarithm
$\ln(e^x)=x$

$( \log_bM=c ) = b^cM$

$\log_a(a)=1$

Example:
$\log_3(3) = 1$

$\log_bM=\frac{\log_aM}{\log_ab}$

Exponential Growth Model

$A=A_0e^{kt}$

$t=$number of years after initial year
$A_0=$population when $t=0$

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