Lesson 9 Exponential and Logarithmic Functions

9.1 Lecture Content

9.2 Lecture Notes

9.2.1 Rules for Exponents

Let $a > 0$, and $m, n \in \mathbb{R}$:

$a^m \cdot a^n = a^{m+n}$
$\frac{a^m}{a^n} = a^{m-n}$
$(a^m)^n = a^{mn}$
$(ab)^n = a^n b^n$
$a^0 = 1$
$a^{-n} = \frac{1}{a^n}$

9.2.2 Rules for Logarithms

Let $a > 0$, $a \neq 1$, and $x, y > 0$:

$\log_a(xy) = \log_a(x) + \log_a(y)$
$\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$
$\log_a(x^n) = n \cdot \log_a(x)$
$\log_a(a) = 1$
$\log_a(1) = 0$

9.2.3 Finding an Exponential Equation from Two Points

To find $f(x) = C \cdot a^x$ that passes through points $(x_1, y_1)$ and $(x_2, y_2)$:

Set up the system: \[ y_1 = C \cdot a^{x_1} \\ y_2 = C \cdot a^{x_2} \]
Divide the two equations to eliminate $C$: \[ \frac{y_2}{y_1} = \frac{a^{x_2}}{a^{x_1}} = a^{x_2 - x_1} \]
Solve for $a$: \[ a = \left( \frac{y_2}{y_1} \right)^{\frac{1}{x_2 - x_1}} \]
Plug back into one equation to solve for $C$.

9.2.4 Solving Exponential and Logarithmic Equations

Exponential Equations

To solve $a^{f(x)} = b$:

Take logarithms of both sides: \[ \log(a^{f(x)}) = \log(b) \]
Use the power rule: \[ f(x) \cdot \log(a) = \log(b) \]
Solve for $x$.

Logarithmic Equations

To solve $\log_a(f(x)) = b$:

Rewrite in exponential form: \[ f(x) = a^b \]
Solve the resulting algebraic equation for $x$.

9.2.5 Applications and Formulas

Continuous Compounding

If interest is compounded continuously:

\[ A = Pe^{rt} \]

Where: - $A$ = final amount - $P$ = principal - $r$ = annual interest rate - $t$ = time in years - $e \approx 2.718$

Total Revenue from Demand Function

If price $p(x)$ depends on demand $x$ (e.g. $p(x) = 120 - 3x$), then:

\[ R(x) = x \cdot p(x) \]

Where: - $R(x)$ = total revenue - $x$ = quantity sold - $p(x)$ = price per unit as a function of demand

You can expand and simplify to analyze maximum revenue using calculus or vertex form (if quadratic).

9.3 Examples

9.3.1 Quadratic Functions

Video: Solutions

Identify $a$, $b$, and $c$ in the equation below, then solve using the Quadratic Formula. \[ x^2 + 18x + 80 = 0 \]
Determine the number and type of solutions for the equation: \[ -2x^2 -9x -9 = 0 \]
The equation \[ 5x^2 +15x +4 = 0 \] has two solutions. Find them.
Consider the quadratic function: \[ f(x) = 9x^2 - 1 \] Find the:
- Vertex
- Largest $x$-intercept
- $y$-intercept
Consider the parabola given by: \[ f(x) = -4x^2 -8x +14 \] Answer the following:
1. The graph of this function opens: ☐ Up ☐ Down
2. Domain (interval notation):
3. Range (interval notation):
The Acme Widget Company has found that if widgets are priced at $218, then $14,000$ will be sold. For every increase of $13, there will be 300 fewer widgets sold. The marginal cost is $130.80 per widget. Fixed costs are $12,000. If $x$ represents the price of a widget, find:
1. Number of widgets sold
2. Revenue
3. Cost of production
4. Profit
5. Price that maximizes profit

9.3.2 Exponent Rules

Video: Solutions

Simplify the expression completely:

\[ \frac{x^{22} \cdot y^{79}}{x^{16} \cdot y^{20}} \]

Simplify the given expression:

\[ \frac{(a^4 z^2)^4 \cdot (-a^2 z^2)^6}{\left( -a z^3 a^5\right)^2} \]

Sketch each of the following:

$4(0.65)^x$
$3(1.17)^x$
$4(1.17)^x$
$4(1.52)^x$
$4(0.88)^x$

Use the like-bases property and exponents to solve:

\[ \left( \frac{1}{5} \right)^{x+7} = 5^{7x+4} \]

The population of the world in 1987 was 5 billion and the annual growth rate was estimated at 2% per year. Assuming an exponential growth model, find the projected world population in 1992.

9.3.3 Logarithm Rules

Video: Solutions

Express the equation in exponential form.

\[ \log_{2} 8 = 3 \]
That is, write your answer in the form $2^A = B$.
\[ \log_{5} 3125 = 5 \]
That is, write your answer in the form $5^C = D$.

Simplify:
\[ \log_{z} \left( \left( \frac{1}{z} \right)^{6} \right) \]
If $\ln(a) = 2$, $\ln(b) = 3$, and $\ln(c) = 5$, evaluate the following:
1. \[ \ln\left( a^{a} b^{-2} c^{-2} \right) \]
2. \[ \ln\left( \sqrt{b^1 c^{-2}a^{-4}} \right) \]
3. \[ \frac{\ln\left( a^{-4} b^{4} \right)}{\ln\left( \frac{b}{c} \right)^{2}} \]
4. \[ \left( \ln c^{-2}\right) \left( \ln \frac{a}{b^3}\right)^{4} \]
Find the domain of: \[ y = \log(5 - 6x) \]
If $e^{7x} = 27$, then $x$ is what?
Solve for $m$ in the equation below. \[ 2 \log_{8}(m) + 4 = 8 \]
Solve for $x$:

\[ \log(x+5) - \log(x+3) = 1 \]

A culture of bacteria grows according to the continuous growth model: \[ B = f(t) = 500 e^{0.073t} \]
where $B$ is the number of bacteria and $t$ is in hours.
1. Find $f(0)$
2. To the nearest whole number, find the number of bacteria after 5 hours.
3. To the nearest tenth of an hour, determine how long it will take for the population to grow to 1100 bacteria.

9.4 Practice Problems

Find the equation of the exponential function that passes through $(2,-4)$ and $(4,-16)$.
Solve $4^x = 3$.
Solve $4(1.5^{2x-1}) = 8$.
Solve $e^{2x} - 9e^x + 20 = 0$.
Why is the logarithm of a negative number not defined?
Sketch $f(x) = \log_5 x$ and $g(x) = 5^x$.
Suppose $\log(a) = 5$, $\log(b) = 3$ and $\log(c) = -2$. What is $\log\left(\dfrac{ab^4}{c^7} \right)$?
(Applied) How long will it take a $500 investment to be worth $700 if it is continuously compounded at 15% per year? (Give the answer to two decimal places.)
(Applied) The Better Baby Buggy Co. has just come out with a new model, the Turbo. The market research department predicts that the demand equation for Turbos is given by \[ q = -2p + 320, \]

where $q$ is the number of buggies it can sell in a month if the price is $p per buggy. At what price should it sell the buggies to get the largest revenue? What is the largest monthly revenue?

9.5 Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

Find the equation of the exponential function that passes through $(3,5)$ and $(4,25)$.
Solve $6^{3x+1} = 30$.
Suppose $\log(a) = 2$, $\log(b) = -7$ and $\log(c) = 3$. What is $\log\left(\dfrac{a^2b^3}{c^5} \right)$?
The median price of a home in the United States declined continuously over the period 2005–2008 at a rate of 5.5% per year from around $230 thousand in 2005. Write down a formula that predicts the median price of a home $t$ years after 2005. Use your model to estimate the median home price in 2007 and 2010.
In 2005, the Las Vegas monorail charged $3 per ride and had an average ridership of about 28,000 per day. In December 2005 the Las Vegas Monorail Company raised the fare to $5 per ride, and average ridership in 2006 plunged to around 19,000 per day.
1. Use the given information to find a linear demand equation.
2. Find the price the company should have charged to maximize revenue from ridership. What is the corresponding daily revenue?
3. The Las Vegas Monorail Company would have needed $44.9 million in revenues from ridership to break even in 2006. Would it have been possible to break even in 2006 by charging a suitable price?

9.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill	D	CON	COM
Simplify expressions involving exponents or logarithms.
Find the equation of an exponential function through two points.
Solve exponential or logarithmic functions.
Maximize the total revenue function, given a linear demand curve.
Solve problems involving the continuous compounding formula.
Solve applied problems involving exponentials or logarithms.