📝 Chapter 6: Exponential & Logarithmic Functions

Convert: a) 4³=64 to log form b) log₇(343)=? c) Evaluate log(10000)

Evaluate: a) log₃(1/27) b) log₈(2) c) ln(1/e²)

Sketch y = log₃(x+2) - 4. State domain, range, x-intercept, and asymptote.

Expand using log laws: log₂(16x⁴y/z³)

Condense to a single logarithm: 2log(x) + (1/2)log(y) - 3log(z)

Without a calculator, determine which is larger: log₂(30) or log₃(50). Show your reasoning.

Prove: logₐ(x) = logᵦ(x)/logᵦ(a) (the change of base formula). Start from the definition of logarithm.

If log(2) ≈ 0.301 and log(3) ≈ 0.477, find the value of log(72) without a calculator. Show all steps.

A student simplifies log(x+y) as log(x) + log(y). Create a numerical example showing this is incorrect. Then explain the correct rule and when addition of logs does apply.

Explain the relationship between exponential functions and logarithmic functions. Include: inverse relationship, how graphs are related (reflection), and how this helps solve equations.

Create a summary of all three logarithm laws (product, quotient, power) with: the rule, a proof or justification, and two examples for each.

Explain why log(-5) is undefined in real numbers. What does this tell us about the domain of logarithmic functions?

Describe how logarithmic transformations (y = a·logₖ(b(x-h)) + v) affect the parent function. Compare to transformations of polynomial functions.

The Richter scale measures earthquake intensity: M = log(I/I₀). An earthquake measures 6.0. a) How many times more intense is it than a 4.0 earthquake? b) A new earthquake is 1000× more intense than the 6.0. What is its Richter rating?

Sound intensity is measured in decibels: dB = 10·log(I/I₀). Normal conversation is 60 dB. A rock concert is 120 dB. How many times more intense is the concert than conversation?

The pH of a solution is defined as pH = -log[H⁺]. a) If [H⁺] = 3.2 × 10⁻⁴, find the pH. b) If pH = 8.5, find [H⁺]. c) How many times more acidic is pH 3 than pH 5?

A painting purchased for $500 appreciates in value according to V(t) = 500·(1.08)ᵗ. a) Express t in terms of V using logarithms. b) How long until the painting is worth $2000? c) What is the average rate of change of value from year 5 to year 10?