Exponential Equations and Graphs

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Questions and Answers

In the equation f(x) = 3^x what is the domain?

All integers, {x ∈ R}

In the equation f(x) = 3^x what is the range?

Only the positive outputs, {y ∈ R | y>0}

In the equation f(x) = 3^x what is the x-intercept?

None

In the equation f(x) = 3^x, what is the zero?

<p>None</p> Signup and view all the answers

In the equation f(x) = 3^x, what is the horizontal asymptote?

<p>y=0</p> Signup and view all the answers

In the equation f(x) = 3^x, what is the vertical asymptote?

<p>None</p> Signup and view all the answers

If we have a bacterial population that doubles every hour and the initial population is 10,000, how many hours will it take to exceed 640,000?

<p>6 hours</p> Signup and view all the answers

The predicted population of LUNHS is given by P = 1000e^(0.10y) where y represents the number of years after 2024. What will be the population in the year 2034?

<p>2,718</p> Signup and view all the answers

What is the half-life of a certain radioactive substance if its initial amount is 20 grams and the amount remaining after 2 minutes (120 seconds) is 2.5 grams?

<p>40 seconds</p> Signup and view all the answers

What is the first step in solving logarithmic inequality such as (x - 2) > 5?

<p>Add 2 to both sides</p> Signup and view all the answers

If the decibel level of sound in a quiet office is 10^-6 watts/m², what is the corresponding sound intensity in decibels?

<p>60 decibels</p> Signup and view all the answers

A 1-liter solution contains 10^-5 moles of hydrogen ions. What is the pH level?

<p>5</p> Signup and view all the answers

What is the domain of the logarithmic function f(x) = logb(x)?

<p>Set of all possible input values</p> Signup and view all the answers

What is the range of the logarithmic function f(x) = logb(x)?

<p>Set of all real numbers</p> Signup and view all the answers

What is the horizontal asymptote of the logarithmic function f(x) = logb(x)?

<p>None</p> Signup and view all the answers

What is the vertical asymptote of the logarithmic function f(x) = logb(x)?

<p>x = 0</p> Signup and view all the answers

Suppose an earthquake released approximately 10^12 joules of energy. What is its magnitude on the Richter scale?

<p>5.1</p> Signup and view all the answers

If a function is the inverse of another function, what is the relationship between their graphs?

<p>They are reflections of each other across the line y = x</p> Signup and view all the answers

In general, how do logarithmic and exponential functions relate to each other?

<p>They are inverses of each other.</p> Signup and view all the answers

Flashcards

Exponential Growth

A process where a quantity increases by a fixed multiple over equal intervals of time.

Compound Interest

Interest earned not only on the principal amount, but also on previously accumulated interest.

Population Growth

The increase in the number of individuals in a population over time.

Exponential Function

A function of the form f(x) = b^x, where 'b' is a positive constant (base) and 'x' is the independent variable.

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Exponential Equation

An equation where the unknown variable is in the exponent.

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Exponential Inequality

An inequality where the unknown variable is in the exponent.

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Solving Exponential Equation

The process of finding the value of the variable in an exponential equation, requiring same bases.

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Solving Exponential Inequality

Finding the solution range of the variable in an exponential inequality.

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Graph of Exponential Function

A curve representing an exponential function.

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Domain of an Exponential Function

All real numbers.

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Range of an Exponential Function

All positive real numbers.

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X-intercept of Exponential Function

There is no x-intercept.

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Y-intercept of Exponential Function

(0,1)

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Horizontal Asymptote of Exponential Function

y = 0

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Logarithmic Function

Inverse of exponential function, y = logb x

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Converting between Logarithm and Exponential

Ability to translate equations between logarithmic and exponential forms.

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Logarithm Properties

Rules for simplifying or combining logarithms (Product and Quotient).

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Graph of Logarithmic Function

Curve representing a logarithmic function.

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Domain of Logarithmic Function

All positive real numbers.

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Range of Logarithmic Function

All real numbers.

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X intercept of Logarithmic Function

(1, 0)

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Y intercept of Logarithmic Function

None

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Solving Logarithmic Equation

Finding the solution for the variable in a logarithmic equation, often converted to exponential form.

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Study Notes

Exponential Equations and Inequalities

  • Exponential growth examples include bacterial growth, compound interest, and population growth.
  • Exponential functions typically have the form f(x) = bx, where 'b' is the base, and cannot be negative. Example: f(x) = 3x
  • Exponential equations, like 3x+1 = 24, involve solving for 'x' when the base is the same for both sides. Example: 3x+2 = 27 becomes 3x+2 = 33, then x+2 = 3 solving for x.
  • When solving inequalities, be cautious when dividing both sides by a negative number – the inequality sign flips.
  • Example: 53x > 52(3x-2)

Graph of Exponential Function

  • Example function: f(x) = 2x
  • X values: -2, -1, 0, 1, 2
  • Corresponding f(x) values: 1/4, 1/2, 1, 2, 4
  • Note graphs of exponential functions start at (0, 1) and increase as x increases.

Exponential Function

  • Domain: All real numbers (x∈ℝ)
  • Range: Positive outputs only (y ∈ℝ| y>0)
  • X-intercept: None
  • Y-intercept: (0, 1)
  • Zero: None
  • Horizontal asymptote: y = 0

Applications

  • A population of bacteria doubles every hour. With initial population of 10,000 bacteria, how many hours for the population to exceed 640,000? (Answer: 6 hours)
  • Predict the population for a given year (year 2034), given a population model (P = 1000 e0.10y, where y = years after 2024). (Answer: 2718)

Solving Logarithmic Equations

  • Example: log3(x - 2) = 3 can be solved like this: 33 = x - 2 giving x = 29

Solving Logarithmic Inequalities

  • Example: log3(x - 3) > 2 can be solved by converting to exponential form: 32 > x - 3 resulting in x < 12

Properties of Logarithms

  • Product property: logb(a * c) = logba + logbc
  • Quotient property: logb(a/c) = logba - logbc

Graph of Logarithmic Function

  • Domain: Positive real numbers (x ∈ℝ| x>0)
  • Range: All real numbers (y ∈ℝ)
  • X-intercept: (1, 0)
  • Y-intercept: None
  • Zero: x = 1
  • Vertical asymptote: x = 0

Applications of Logarithms

  • Finding earthquake magnitude (Richter scale)
  • Determining acidity levels (pH)
  • Calculating sound intensity (decibels)

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