Tasks 02-05 - Exponential, Logarithmic & Complex Word Problems

Section 02: Equations & Problem-Solving Strategies

Instructions

Complete these problems to master advanced exponential and logarithmic equations, as well as complex multi-step word problems. These problems integrate all equation-solving techniques from Section 02.

Problem 1: Exponential Equations (x)

Solve each exponential equation:

  1. \(2^{x+3} = 128\)

  2. \(5^x \cdot 25^{x-1} = 125\)

  3. \(3^{2x} - 12 \cdot 3^x + 27 = 0\)

  4. \(4^x - 2^{x+1} - 8 = 0\)

Problem 2: Logarithmic Equations (x)

Solve each logarithmic equation, stating domain restrictions:

  1. \(\log_3(x + 4) = 2\)

  2. \(\log(x) + \log(x + 3) = 1\)

  3. \(\log_2(x) - \log_8(x) = 1\)

  4. \(\log_x(16) = 4\)

Problem 3: Mixed Exponential-Logarithmic (xx)

Solve these equations that involve both exponential and logarithmic expressions:

  1. \(3^{\log_3(x)} + x = 10\)

  2. \(\log_2(2^x + 1) = x - 1\)

Note, there might be multiple solutions or no solution at all!

Problem 4: Growth and Decay Applications (xx)

A radioactive substance decays according to \(A(t) = A_0 e^{-kt}\) where \(t\) is in years.

  1. If 70% remains after 5 years, find the decay constant \(k\)
  2. Find the half-life of the substance
  3. How long until only 10% remains?
  4. If we start with 100 grams, when will exactly 25 grams remain?

Problem 5: Investment Comparison (xx)

An investor has €30,000 to invest. She’s considering three options:

  • Option A: 6% annual interest, compounded yearly
  • Option B: 5.8% annual interest, compounded monthly
  • Option C: 5.7% annual interest, compounded continuously
  1. Write the formula for each option after \(t\) years
  2. Which option yields the most after 10 years?
  3. If she wants €50,000, how long would each option take?

Problem 6: Population Dynamics (xxx)

Two bacterial cultures are growing in a lab. Culture A starts with 500 bacteria and doubles every 3 hours. Culture B starts with 800 bacteria and grows according to \(B(t) = 800e^{0.15t}\) where \(t\) is in hours.

  1. Write the growth equation for Culture A
  2. When will the populations be equal?