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Quizzes > High School Quizzes > Mathematics

6.08 Exponential Equations Practice Quiz

Master exponential applications with engaging quiz tests

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art representing a trivia quiz for high school algebra students on solving exponent equations.

Solve the equation 2^x = 16. What is x?
x = 5
x = 2
x = 3
x = 4
16 can be written as 2^4. Since the bases are the same, the exponents must be equal; therefore, x = 4.
What is the value of 10^0?
Undefined
0
10
1
Any nonzero number raised to the power of 0 equals 1. This is a fundamental property of exponents.
Solve the equation 3^(x-1) = 9 for x.
4
2
1
3
Since 9 equals 3^2, the equation becomes 3^(x-1) = 3^2. Equating the exponents gives x - 1 = 2, so x = 3.
Simplify the expression 2^3 * 2^2.
2^6
2^5
2^4
2^3
When multiplying powers with the same base, you add the exponents. Thus, 2^3 * 2^2 = 2^(3+2) = 2^5.
Express 81 as a power of 3.
3^4
3^5
3^3
3^2
Since 3 multiplied by itself four times equals 81, we can write 81 as 3^4. This is a direct application of exponentiation.
Solve the equation 4^x = 64 for x.
6
2
3
4
Recognize that 64 is a power of 4 since 4^3 = 64. Equating the exponents leads directly to x = 3.
Solve the exponential equation 5^(x+1) = 125.
4
1
3
2
Since 125 equals 5^3, we set x + 1 equal to 3. Solving this gives x = 2.
Solve the equation 2^(2x) = 8 for x.
3
2
3/2
1
Since 8 can be expressed as 2^3, the equation becomes 2^(2x) = 2^3. Equating exponents gives 2x = 3, resulting in x = 3/2.
Calculate x if (1/2)^x = 8.
3
-8
-3
0
Rewrite (1/2)^x as 2^(-x) and note that 8 is 2^3. Setting the exponents equal gives -x = 3, so x = -3.
Solve for x in the equation 9^(x-1) = 27.
5/2
7/2
2
3
Express 9 as 3^2 and 27 as 3^3 so the equation becomes 3^(2x-2) = 3^3. Equating the exponents yields 2x - 2 = 3, hence x = 5/2.
Find x if 2^(3x) = 64.
1
6
2
3
Since 64 can be written as 2^6, equate 3x to 6. Dividing both sides by 3 gives x = 2.
Determine x such that 3^(2x) = 81.
2
4
3
1
Recognize that 81 is 3^4. Equate the exponents in 3^(2x) = 3^4 to obtain 2x = 4, so x = 2.
Solve the equation 7^(x+2) = 49 * 7^x for x.
All real numbers
x = 0
x = -2
x = 2
Since 49 is 7^2, the right side can be rewritten as 7^(x+2), making the equation an identity. This means the equation holds for any real value of x.
Solve the equation e^x = e^5 for x.
1
0
e^5
5
Because the exponential function is one-to-one, e^x = e^5 implies x = 5. This directly uses the property of exponential functions.
Find x if 10^(2x-1) = 1.
0
-0.5
0.5
1
Since 1 can be written as 10^0, we set the exponent 2x - 1 equal to 0. Solving 2x - 1 = 0 gives x = 0.5.
Solve the equation 2^(x+1) = 3^(2x-2) for x. Express your answer in terms of logarithms.
x = (ln3 + ln2) / (ln3 - ln2)
x = (2 ln2 + ln3) / (2 ln2 - ln3)
x = (2 ln3 - ln2) / (2 ln3 + ln2)
x = (2 ln3 + ln2) / (2 ln3 - ln2)
Taking the natural logarithm on both sides lets you bring the exponents down as coefficients. After rearranging and isolating x, the solution simplifies to x = (2 ln3 + ln2) / (2 ln3 - ln2).
Solve the equation 10^x = 2^(x+1) * 5^(x-1) for x.
x = 0
No solution
x = -1
x = 1
Rewriting the right-hand side in terms of 10^x shows that it becomes (2/5) * 10^x. Dividing both sides by 10^x leads to the false statement 1 = 2/5, indicating there is no solution.
Solve the equation 8^x = 5^(x+1) for x using logarithms.
x = ln5 / (ln8 - ln5)
x = ln8 / (ln5 - ln8)
x = (ln8 - ln5) / ln5
x = ln5 / (ln5 - ln8)
Taking logarithms of both sides allows you to bring down the exponents. Rearranging the equation yields x = ln5 / (ln8 - ln5), which is the correct solution.
Solve the equation 4^(2x) = 8^(x+3) for x.
6
12
3
9
Write 4 as 2^2 and 8 as 2^3; then the equation becomes 2^(4x) = 2^(3x+9). Equating the exponents, 4x = 3x + 9, leads to the solution x = 9.
Find x such that 3^(2x+1) = 2 * 9^x.
x = 1
x = 2
No solution
x = 0
First, express 9^x as (3^2)^x = 3^(2x). The equation then becomes 3^(2x+1) = 2 * 3^(2x). Dividing both sides by 3^(2x) yields 3 = 2, a contradiction that means there is no solution.
0
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Study Outcomes

  1. Solve various exponential equations using established exponent rules.
  2. Apply the laws of exponents to simplify and transform expressions.
  3. Analyze structures of exponential equations to identify solution strategies.
  4. Evaluate and verify solutions through substitution and comparison.

6.08 Quiz - Exponential Equation Apps Cheat Sheet

  1. Master Exponent Rules - Get comfy with the product, quotient, and power rules that let you simplify exponent expressions like a breeze. These rules are your toolkit for dismantling and solving exponential equations deftly. Once you've got them down, complex problems start to feel like child's play! Practice Worksheet at GeeksforGeeks
    2. https://www.geeksforgeeks.org/exponential-equations-worksheet/
  2. Rewrite with Common Bases - Turn tricky equations into simple puzzles by expressing both sides of the equation with the same base. When the bases match, you can equate exponents directly and breeze through the solution. This technique saves time and reduces errors. Coreq Algebra Guide at Symbolab
    3. https://www.symbolab.com/study-guides/coreq-collegealgebra/exponential-equations.html
  3. Use Logarithms for Different Bases - When two sides refuse to share a base, invite logarithms to the party. Applying log to both sides isolates the exponent, turning a fiddly equation into a straightforward solve. It's a powerful trick you'll pull out again and again! Log Guide at Symbolab
    4. https://www.symbolab.com/study-guides/coreq-collegealgebra/exponential-equations.html
  4. Watch for Extraneous Solutions - Exponential algebra loves sneaking in bogus answers when you manipulate both sides. Always plug your solutions back into the original equation to confirm they actually work. There's no shame in checking your work! Extraneous Solution Tips at Symbolab
    5. https://www.symbolab.com/study-guides/collegealgebracoreq/exponential-equations-with-like-bases.html
  5. Explore Real‑World Models - Exponential equations pop up in population growth, radioactive decay, and more - unlocking these applications makes math feel alive! Working through real data helps cement your skills and shows why you're learning this magic. It's like unlocking the secret code of the universe. Real‑World Worksheet at GeeksforGeeks
    6. https://www.geeksforgeeks.org/exponential-equations-worksheet/
  6. Handle Fractional Exponents - Turn roots into rational exponents by flipping the script: √x becomes x^(1/2), etc. This rewrite makes it easy to apply your exponent rules and solve swiftly. Break it down step‑by‑step and watch complexity melt away! Fractional Exponent Practice at Symbolab
    7. https://www.symbolab.com/study-guides/coreq-collegealgebra/exponential-equations.html
  7. Substitute to Form Quadratics - Some exponential equations hide a quadratic in disguise - spot it by setting a = base^x. You'll transform the equation into something you can factor or use the quadratic formula on. This clever detour opens up fresh solving strategies. Quadratic‑Style Tips at Symbolab
    8. https://www.symbolab.com/study-guides/collegealgebracoreq/exponential-equations-with-like-bases.html
  8. Tackle Natural Exponentials - Equations with the big E (e) call for natural logs (ln) to unravel continuous growth or decay models. Once you apply ln, the exponent steps down and you can solve like any other linear term. Perfect for biology, finance, or physics applications! Natural Expo Guide at Symbolab
    9. https://www.symbolab.com/study-guides/coreq-collegealgebra/exponential-equations.html
  9. Simplify Negative Exponents - Rewrite negative exponents as fractions (a❻❿ = 1/a❿) to clean up the equation and see the solution path clearly. This simple flip turns intimidating terms into friendly fractions you can handle. Practice this move until it's second nature! Negative Expo Drills at GeeksforGeeks
    10. https://www.geeksforgeeks.org/exponential-equations-worksheet/
  10. Build a Problem‑Solving Toolbox - The ultimate cheat code? Variety! Tackle different problem types - straightforward, real‑world, and edge‑case equations - to sharpen your intuition. Challenge yourself with timed drills and mix strategies to stay on your A‑game. Practice and Mastery at MathBits
    11. https://mathbitsnotebook.com/Algebra1/Exponentials/EXEquationPRactice.html
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