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Unit 3: Exponential and Logarithmic Functions

Growth and decay built from a constant multiplier, the natural base e, and logarithms as the inverse operation that undoes an exponent.

11 topics · 41 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Exponential Growth and Decay

  1. How it worksHow does a constant growth factor multiply a quantity again and again to produce exponential growth?
  2. Worked exampleFind the value after 5 years of a $1000 investment growing at 6 percent annually, compounded once a year.
  3. Worked exampleFind the amount remaining after 3 years of a 200 mg sample decaying at 15 percent per year.
  4. Common mistakeWhy isn't exponential growth the same as growth that just keeps speeding up by a fixed amount?
Your turn1 of 2
Write the value after t years of a $3634 investment growing at 7 percent annually, compounded once a year, as a function of t.

The Natural Base e

  1. How it worksHow does compounding interest more and more often push the growth factor toward the number e?
  2. Worked exampleFind the value of $500 after 3 years at 8 percent interest compounded continuously.
  3. Common mistakeWhy isn't e just an approximation that mathematicians rounded to for convenience?
Your turn1 of 2
Write the value of $1423 after t years at 8 percent interest compounded continuously, as a function of t.

Logarithms as Inverses of Exponentials

  1. How it worksHow does log base b of x answer the question of what exponent turns b into x?
  2. Worked exampleFind log base 2 of 32 and log base 5 of 1/25.
  3. Common mistakeWhy isn't log(x) the same operation as dividing by x?
  4. Ties togetherHow does a logarithm undoing an exponential connect to a graph reflecting across the line y = x?
Your turn1 of 2
Find log base 5 of 54.

Graphs of Logarithmic Functions

  1. How it worksHow does reflecting an exponential graph across the line y = x produce the logarithmic graph?
  2. Worked exampleFind the domain, range, and vertical asymptote of f(x) = log base 3 of (x - 2).
  3. Common mistakeWhy can't a logarithmic function ever have zero or a negative number in its domain?
Your turn1 of 2
Find the vertical asymptote's x-value for f(x) = log base 5 of (x - 6).

Properties of Logarithms

  1. How it worksHow do the product, quotient, and power properties let you rewrite a single complicated logarithm?
  2. Worked exampleFind log(50) given that log(2) is about 0.301 and log(25) is about 1.398.
  3. Common mistakeWhy isn't log(x+y) the same as log(x) + log(y)?
  4. Common mistakeWhy isn't log(x2) the same as (log(x))2?
Your turn1 of 2
Find the expanded form of log(x3 / y3).

Expanding and Condensing Logarithmic Expressions

  1. How it worksHow do you expand log((x2·y)/z3) into a sum and difference of simpler logarithms?
  2. Worked exampleFind the expanded form of log((x3*√y)/z).
  3. Worked exampleFind the condensed single logarithm equal to 2 log(x) + log(y) - 3 log(z).
  4. Common mistakeWhy can't you split log(x + y) into log(x) + log(y) the way you can split log(xy)?
Your turn1 of 2
Find the expanded form of log((x3·y)/z3).

The Change of Base Formula

  1. How it worksHow does the change of base formula rewrite any logarithm using only base 10 or base e?
  2. Worked exampleFind log base 5 of 12 using the change of base formula and a calculator's base-10 log.
  3. Common mistakeWhy doesn't changing the base of a logarithm change the actual value it represents?
Your turn1 of 2
Write log base 7 of x using the change of base formula, as a function of x.

Solving Exponential Equations

  1. How it worksHow does taking the logarithm of both sides pull a variable down out of the exponent?
  2. Worked exampleFind x in the equation 3x+1 = 20 using logarithms.
  3. Worked exampleFind x in the equation 2x = 8x-3 by matching bases.
  4. Common mistakeWhy can't you just divide both sides of 2x = 10 by 2 to solve for x?
Your turn1 of 2
Find x in the equation 5x+3 = 94 using logarithms.

Solving Logarithmic Equations

  1. How it worksHow does exponentiating both sides of a logarithmic equation undo the logarithm?
  2. Worked exampleFind x in the equation log(x) + log(x-3) = 1.
  3. Common mistakeWhy does an extraneous solution show up when solving log(x) + log(x-3) = 1, and how do you catch it?
  4. Ties togetherHow does exponentiating to undo a logarithm mirror taking logs to undo an exponential?
Your turn1 of 2
Find x in the equation log(x) + log(x - 4) = 1.47.

Compound Interest and Continuous Growth Models

  1. How it worksHow does the compound interest formula A = P(1+r/n)nt account for compounding frequency?
  2. Worked exampleFind the balance after 10 years on a $2000 deposit at 5 percent interest compounded quarterly.
  3. Worked exampleFind how long it takes $1000 to double at 4 percent interest compounded continuously.
  4. Going furtherWhat happens to the final balance as the number of compounding periods per year grows without bound?
Your turn1 of 2
Write the balance after t years on a $3634 deposit at 7 percent interest compounded 11 times per year, as a function of t.

Logistic Growth Models

  1. How it worksHow does a carrying capacity cap an S-shaped logistic curve after its early exponential-looking rise?
  2. Worked exampleFind the population predicted by a logistic model with capacity 1000, growth rate 0.3, and an early value of 50 at t = 10.
  3. Common mistakeWhy isn't logistic growth just exponential growth that happens to stop suddenly?
  4. Ties togetherHow does the early rise of a logistic curve reduce to the plain exponential growth from the start of this unit?
Your turn1 of 2
A logistic model has carrying capacity 1545, growth rate 0.36, and initial population 94 at t=0. Write the predicted population as a function of t.