Logarithms Solve Exponential

Grade 11 · algebra · 82 practice problems · read aloud

🔊 Listen to this explanation

Solving Exponential Equations with Logarithms

🔍 What is it and Why is it Useful?

Logarithms are the inverse operation of exponentiation. They are essential for solving equations where the variable is in the exponent (e.g., \( 3^x = 7 \)). Without logs, these equations are often impossible to solve. This skill is crucial in fields like finance (compound interest), science (radioactive decay), and computer science.

📝 Step-by-Step Guide

  1. Isolate the exponential expression on one side of the equation.
  2. Take the logarithm of both sides. You can use any base, but log (base 10) or ln (base \( e \)) are most common on calculators.
  3. Apply the Power Rule: \( \log(b^x) = x \cdot \log(b) \). This brings the exponent down in front.
  4. Solve for the variable using standard algebraic techniques.
  5. Check your solution if possible by substituting it back into the original equation.

✨ Visual Examples

Example 1: Solve \( 4^{x} = 16 \)**

  1. The exponential \( 4^x \) is already isolated.
  2. Take the log of both sides: \( \log(4^x) = \log(16) \).
  3. Use the Power Rule: \( x \cdot \log(4) = \log(16) \).
  4. Solve for \( x \): \( x = \frac{\log(16)}{\log(4)} \).
  5. Since 16 is \( 4^2 \), we know \( x = 2 \).

Example 2: Solve \( 5^{2x-1} = 125 \)**

  1. The exponential is isolated.
  2. Take the log of both sides: \( \log(5^{2x-1}) = \log(125) \).
  3. Use the Power Rule: \( (2x - 1)\log(5) = \log(125) \).
  4. Recognize 125 is \( 5^3 \), so \( (2x - 1)\log(5) = 3\log(5) \).
  5. Divide both sides by \( \log(5) \): \( 2x - 1 = 3 \).
  6. Solve: \( 2x = 4 \), so \( x = 2 \).

⚠️ Common Mistakes

  • Misapplying Log Rules: \( \log(a + b) \) is NOT \( \log(a) + \log(b) \). This only works for multiplication inside the log.
  • Incorrect Isolation: Failing to completely isolate the exponential term before taking the log.
  • Calculator Errors: Forgetting parentheses when entering expressions like \( \frac{\log(17)}{\log(5)} \) into the calculator.
  • Forgetting the Base: When you see \( \log(x) \), remember it has a base of 10. \( \ln(x) \) has a base of \( e \).

💡 Tips & Tricks

  • Base Matching: Always check if you can express both sides with the same base first. If \( 8^x = 32 \), write as \( (2^3)^x = 2^5 \), which is easier than using logs.
  • Consistency is Key: Use the same base for your logarithm on both sides of the equation.
  • Estimate: Before solving \( 10^x = 750 \), estimate: \( 10^2=100 \) and \( 10^3=1000 \), so \( x \) should be between 2 and 3.

🎯 Practice Suggestions

To master this skill, practice with progressively harder problems:

  1. Start with problems that have the same base (e.g., \( 2^x = 8 \)).
  2. Move to problems requiring a single log application (e.g., \( 3^x = 20 \)).
  3. Tackle problems where the exponent is an expression (e.g., \( 4^{2x+1} = 9 \)).
  4. Create a "problem bank" and solve 3-5 problems daily for a week to build fluency and speed.

Practice problems

6 of the 82, worked through step by step — try them before opening the answer.

1 4^(2x) = 48

Hint: Take the logarithm of both sides to bring the variable down from the exponent. Remember the power rule: log(a^b) = b log(a). Then isolate x.

Show the answer

Answer: x = (log(48))/(2 log(4)) ≈ 1.396

  1. Take the natural log (or common log) of both sides: ln(4^(2x)) = ln(48).
  2. Apply the power rule: 2x ln(4) = ln(48).
  3. Divide both sides by 2 ln(4): x = ln(48) / (2 ln(4)).
  4. Simplify if desired: ln(48) = ln(16 * 3) = ln(16) + ln(3) = 4 ln(2) + ln(3). Also ln(4) = 2 ln(2). So x = (4 ln(2) + ln(3)) / (4 ln(2)) = 1 + ln(3)/(4 ln(2)).
  5. Approximate: ln(3) ≈ 1.0986, ln(2) ≈ 0.6931, so x ≈ 1 + 1.0986/(4*0.6931) = 1 + 1.0986/2.7724 ≈ 1 + 0.396 = 1.396.

The answer is x = ln(48)/(2 ln(4)) ≈ 1.396.

2 7^(x+4) = 84

Hint: To solve an exponential equation where the variable is in the exponent, take the logarithm of both sides using any base, then use the power rule to bring the exponent down as a coefficient.

Show the answer

Answer: x = (log(84)/log(7)) - 4 ≈ -1.71

  1. Take the natural logarithm (or common logarithm) of both sides: ln(7^(x+4)) = ln(84).
  2. Apply the power rule of logarithms: (x+4) * ln(7) = ln(84).
  3. Divide both sides by ln(7) to isolate (x+4): x+4 = ln(84)/ln(7).
  4. Subtract 4 from both sides: x = (ln(84)/ln(7)) - 4.
  5. Compute using a calculator: ln(84) ≈ 4.4308, ln(7) ≈ 1.9459, so ln(84)/ln(7) ≈ 2.277, then x ≈ 2.277 - 4 = -1.723.

The answer is x = (ln(84)/ln(7)) - 4 ≈ -1.72.

3 6^(x+1) = 216

Hint: Rewrite 216 as a power of 6, then set the exponents equal to each other.

Show the answer

Answer: 2

  1. Recognize that 216 = 6^3.
  2. Rewrite the equation as 6^(x+1) = 6^3.
  3. Since the bases are equal, set the exponents equal: x + 1 = 3.
  4. Subtract 1 from both sides: x = 2.

The answer is 2.

4 4^(x+2) = 128

Hint: Think about how you can rewrite both sides of the equation with the same base, then use logarithms to solve for the variable in the exponent.

Show the answer

Answer: x = 1/2

  1. Rewrite both sides with the same base. 4 = 2^2 and 128 = 2^7.
  2. Substitute: (2^2)^(x+2) = 2^7
  3. Apply power rule: 2^(2(x+2)) = 2^7
  4. Since bases are equal, set exponents equal: 2(x+2) = 7
  5. Expand: 2x + 4 = 7
  6. Subtract 4 from both sides: 2x = 3
  7. Divide by 2: x = 3/2

The answer is x = 3/2.

5 7^(x+2) = 343

Hint: Think about rewriting 343 as a power of 7. Then use the property that if the bases are equal, the exponents must be equal.

Show the answer

Answer: 1

  1. Recognize that 343 = 7^3.
  2. Rewrite the equation as 7^(x+2) = 7^3.
  3. Since the bases are the same, set the exponents equal: x + 2 = 3.
  4. Subtract 2 from both sides: x = 1.

The answer is 1.

6 5^(x+2) = 1250

Hint: Take the logarithm of both sides of the equation to bring the variable out of the exponent. Then use the power rule of logarithms to solve for x.

Show the answer

Answer: x ≈ 2.4307

  1. Start with the equation 5^(x+2) = 1250.
  2. Take the common logarithm (base 10) of both sides: log(5^(x+2)) = log(1250).
  3. Apply the power rule of logarithms: (x+2) * log(5) = log(1250).
  4. Divide both sides by log(5): x+2 = log(1250) / log(5).
  5. Calculate log(1250) and log(5) using a calculator: log(1250) ≈ 3.09691, log(5) ≈ 0.69897.
  6. Divide: 3.09691 / 0.69897 ≈ 4.4307.
  7. Subtract 2 from both sides: x = 4.4307 - 2 = 2.4307.

The answer is x ≈ 2.4307.

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