Exponential Logarithmic Transformations

Grade 12 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

📈 Exponential & Logarithmic Transformations

What is this? These transformations allow us to convert between exponential and logarithmic forms, which is essential for solving equations where the variable appears in exponents.

Why it's useful: Many real-world phenomena (population growth, radioactive decay, compound interest) follow exponential patterns. Logarithms help us solve for unknown exponents and analyze these models.

🔄 The Fundamental Relationship

Core Transformation: \( y = b^x \quad \Leftrightarrow \quad \log_b(y) = x \)

This means: "b raised to what power gives y?" is equivalent to "log base b of y equals x."

📝 Step-by-Step Problem Solving

  1. Identify the form - Is it exponential or logarithmic?
  2. Apply the transformation - Use \( y = b^x \Leftrightarrow \log_b(y) = x \)
  3. Simplify - Use log properties or exponent rules
  4. Solve - Isolate the variable
  5. Check - Verify your solution makes sense

🔢 Worked Examples

Example 1: Exponential to Logarithmic

Problem: Convert \( 5^3 = 125 \) to logarithmic form

Solution: Base = 5, exponent = 3, result = 125

\( 5^3 = 125 \Rightarrow \log_5(125) = 3 \)

Example 2: Solve Exponential Equation

Problem: Solve \( 2^{x+1} = 32 \)

Step 1: \( 2^{x+1} = 2^5 \) (since 32 = 2⁵)

Step 2: \( x + 1 = 5 \) (bases equal ⇒ exponents equal)

Step 3: \( x = 4 \)

Example 3: Solve Logarithmic Equation

Problem: Solve \( \log_3(x - 2) = 4 \)

Step 1: Transform: \( 3^4 = x - 2 \)

Step 2: \( 81 = x - 2 \)

Step 3: \( x = 83 \)

⚠️ Common Mistakes to Avoid

  • Misapplying log properties: \( \log(a + b) eq \log a + \log b \)
  • Forgetting domain restrictions: \( \log_b(x) \) only exists for \( x > 0 \)
  • Base confusion: \( \log x \) usually means \( \log_{10} x \), ln x means \( \log_e x \)
  • Incorrect transformation: \( \log_b(x) = y \Rightarrow b^y = x \), not \( x^y = b \)

💡 Tips & Strategies

  • Memory aid: "The base of the log becomes the base of the power"
  • Check your work: Substitute back into the original equation
  • Know your bases: Common bases: 2, e, 10
  • Use change of base: \( \log_b(a) = \frac{\log a}{\log b} \) for calculator work

🎯 Practice Suggestions

  • Start with basic transformations between forms
  • Practice solving equations with different bases
  • Work with real-world applications (compound interest, pH levels)
  • Create flashcards of log properties and exponent rules
  • Mix practice between exponential and logarithmic problems

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 log₂(8) + ln(e³) = ?

Hint: Remember that logarithms and natural logs have special properties when their arguments are powers of the base. For example, log₁₀(100) equals 2 because 10² = 100.

Show the answer

Answer: 6

  1. Understand the problem We have: log₂(8) + ln(e³)
  2. Evaluate log₂(8) log₂(8) means: "2 raised to what power equals 8?" Since 2³ = 8, log₂(8) = 3.
  3. Evaluate ln(e³) ln means natural logarithm, which is log base e. ln(e³) means: "e raised to what power equals e³?"
  4. Add the results log₂(8) + ln(e³) = 3 + 3 = 6.
  5. Final answer

Let's solve step by step. The answer is 3, because e³ = e³. So ln(e³) = 3. The correct answer is 6.

2 log₂(4) + ln(e³) = ?

Hint: Remember that logarithms and natural logs have special properties when their arguments are powers of the base. For example, log₁₀(100) equals 2 because 10² = 100.

Show the answer

Answer: 5

  1. Understand the problem We need to evaluate: log₂(4) + ln(e³) ---
  2. Evaluate log₂(4) log₂(4) means: "2 to what power equals 4?" Since 2² = 4, log₂(4) = 2 ---
  3. Evaluate ln(e³) ln means natural logarithm (base e). ln(e³) means: "e to what power equals e³?" By the logarithm power rule: ln(e³) = 3 × ln(e) And ln(e) = 1, so ln(e³) = 3 × 1 = 3 ---
  4. Add the results log₂(4) + ln(e³) = 2 + 3 = 5 --- Final answer: 5

3 log₂(16) + ln(e⁴) = ?

Hint: Remember that logarithms and natural logs have special properties when the argument matches the base.

Show the answer

Answer: 8

  1. Evaluate log₂(16). Since 2^4 = 16, log₂(16) = 4.
  2. Evaluate ln(e⁴). Since ln(e^x) = x, ln(e⁴) = 4.
  3. Add the results: 4 + 4 = 8.

The answer is 8.

4 e^(ln(5) + 2ln(3)) = ?

Hint: Remember that e^ln(x) = x and that ln(a^b) = b*ln(a). Try simplifying the expression inside the exponent first.

Show the answer

Answer: 45

  1. Start with e^(ln(5) + 2ln(3))
  2. Use the property ln(a^b) = b*ln(a) to rewrite 2ln(3) as ln(3^2) = ln(9)
  3. Now we have e^(ln(5) + ln(9))
  4. Use the property ln(a) + ln(b) = ln(ab) to combine: ln(5) + ln(9) = ln(5×9) = ln(45)
  5. Now we have e^(ln(45))
  6. Since e^ln(x) = x, e^(ln(45)) = 45

The answer is 45.

5 e^(2ln(3) + ln(2)) = ?

Hint: Remember that e^(ln(x)) = x and use exponent rules to simplify the expression

Show the answer

Answer: 18

  1. Start with e^(2ln(3) + ln(2))
  2. Use exponent rule: 2ln(3) = ln(3^2) = ln(9)
  3. Now we have e^(ln(9) + ln(2))
  4. Use logarithm property: ln(9) + ln(2) = ln(9 × 2) = ln(18)
  5. Now we have e^(ln(18))
  6. Since e^(ln(x)) = x, e^(ln(18)) = 18

The answer is 18.

6 e^(2ln(3) - ln(2)) = ?

Hint: Use properties of exponents and logarithms to simplify the expression before calculating

Show the answer

Answer: 4.5

  1. Apply the property a·ln(b) = ln(b^a) to 2ln(3): 2ln(3) = ln(3^2) = ln(9)
  2. Rewrite the expression: e^(ln(9) - ln(2))
  3. Apply the property ln(a) - ln(b) = ln(a/b): e^(ln(9/2))
  4. Apply the property e^(ln(x)) = x: 9/2
  5. Convert to decimal: 9/2 = 4.5

The answer is 4.5.

Practise this topic — 10 free problems, no signup →