Recap Sheet

Logarithmic Functions & Applications

Logarithmic Functions - Problem Set

Problem 1: Basic Logarithm Evaluation (x)

Evaluate the following logarithms without a calculator:

  1. \(\log_3(81)\)

  2. \(\log_5(125)\)

  3. \(\log_2(\frac{1}{16})\)

  4. \(\log_{10}(0.01)\)

Problem 2: Converting Between Forms (x)

Convert between exponential and logarithmic forms:

  1. Convert to logarithmic form: \(4^3 = 64\)

  2. Convert to logarithmic form: \(10^{-1} = 0.1\)

  3. Convert to exponential form: \(\log_7(343) = 3\)

  4. Convert to exponential form: \(\log_2(32) = 5\)

Problem 3: Using Logarithm Properties (xx)

Simplify the following expressions using logarithm properties:

  1. \(\log_2(16) + \log_2(8)\)

  2. \(\log_5(625) - \log_5(25)\)

  3. \(3\log_4(2)\)

  4. \(\log_3(27x^3) - \log_3(9x)\)

Problem 4: Solving Simple Exponential Equations (xx)

Solve the following exponential equations:

  1. \(3^x = 27\)

  2. \(2^{x-1} = 16\)

  3. \(5 \cdot 2^x = 160\)

  4. \(4^{2x} = 64\)

Problem 5: Solving Exponential Equations with Logarithms (xx)

Solve using logarithms (give exact answers and decimal approximations):

  1. \(2^x = 10\)

  2. \(3^{x+1} = 20\)

  3. \(5^{2x} = 100\)

  4. \(1.08^t = 2\)

Problem 6: pH Scale Application (xx)

The pH of a solution is given by \(\text{pH} = -\log[H^+]\) where \([H^+]\) is the hydrogen ion concentration in moles/liter.

  1. Find the pH of a solution with \([H^+] = 10^{-5}\) moles/liter

  2. Find the pH of a solution with \([H^+] = 10^{-3.5}\) moles/liter

  3. If coffee has pH 5.0, what is its hydrogen ion concentration?

  4. How many times more acidic is a solution with pH 3 than one with pH 6?

Problem 7: Investment and Compound Interest (xxx)

Use logarithms to solve these compound interest problems:

  1. How long will it take €6,000 to grow to €10,000 at 5% annual interest compounded annually?

  2. An investment doubles in 10 years. What is the annual interest rate (compounded annually)?

  3. How long does it take to double your money at 7% annual interest compounded quarterly?

  4. Compare: Which is better - 6% compounded monthly for 10 years or 6.1% compounded annually for 10 years?

Problem 8: Richter Scale and Earthquake Intensity (xxxx)

The Richter scale magnitude is given by \(M = \log(A/A_0)\) where \(A\) is the amplitude and \(A_0\) is a reference amplitude.

  1. An earthquake has magnitude 6.5. Another earthquake has 50 times the amplitude. What is its magnitude?

  2. The 2011 Japan earthquake (magnitude 9.0) was how many times stronger in amplitude than a magnitude 6.0 earthquake?

  3. If earthquake A has magnitude 7.2 and earthquake B has magnitude 5.7, find the ratio of their amplitudes.

  4. Two earthquakes occur simultaneously with magnitudes 6.0 and 6.3. What would be the magnitude if their amplitudes were combined?