Week 1 · Powers of powers and exponential equations: the laws of exponents for real bases and integer exponents, and equations of the form a^x = b · 3 lessons
3
Exponential equations: same base, bases made equal, checking, and word problems
Mathematics Prep 3 Term 1, Unit 1, Lessons 1–2
GoalSolve equations of the form aˣ = b by writing both sides with the same base and equating the exponents, and check the answer.
New words
exponential equation · an equation with the unknown in the exponent, such as 3ˣ = 81same-base property · if a > 0, a ≠ 1 and aˣ = aʸ, then x = yprime factorisation · writing a number as a product of primes, such as 81 = 3⁴check by substitution · putting the answer back into the equation to see both sides agree
Explanation
In an exponential equation the unknown is in the exponent. The key idea: for a positive base other than 1, different exponents give different results (2¹ = 2, 2² = 4, 2³ = 8 …), so if aˣ = aʸ then x = y. We need a > 0 and a ≠ 1 because 1ˣ = 1 for every x, which tells us nothing about x, and a negative base would change sign from one power to the next. The plan: (1) write each number as a power of a small prime, for example 81 = 3⁴, 125 = 5³, 32 = 2⁵, 1 ÷ 9 = 3⁻²; (2) use the laws to make both sides one power of the same base; (3) equate the exponents and solve the ordinary equation; (4) check by substitution. Equations such as 2ˣ = 0 or 3ˣ = −9 have no solution, because a positive base never gives zero or a negative number.
Base
Powers to know (exponents 1, 2, 3, 4, 5, 6)
2
2, 4, 8, 16, 32, 64
3
3, 9, 27, 81, 243, 729
5
5, 25, 125, 625, 3125
10
10, 100, 1000, 10000
Worked examples
Solve 3^(x+1) = 81. Write 81 = 3⁴, so 3^(x+1) = 3⁴. The bases are equal, so x + 1 = 4 and x = 3. Check: 3^(3+1) = 3⁴ = 81. ✓
Solve 16^(x−1) = 64^(x−3) in two ways. With base 4: 16 = 4² and 64 = 4³, so 4^(2x−2) = 4^(3x−9), giving 2x − 2 = 3x − 9 and x = 7. With base 2: 16 = 2⁴ and 64 = 2⁶, so 2^(4x−4) = 2^(6x−18), giving 4x − 4 = 6x − 18, 14 = 2x and x = 7. Both bases work and give the same answer. Check: 16⁶ = 2²⁴ and 64⁴ = 2²⁴. ✓
Solve (2/3)ˣ = 9/4. Since 9/4 = (3/2)² and 3/2 is the reciprocal of 2/3, we have 9/4 = (2/3)⁻². So (2/3)ˣ = (2/3)⁻² and x = −2. Check: (2/3)⁻² = (3/2)² = 9/4. ✓
Class activity
Paper folds. Fold a sheet of paper in half again and again, counting the layers after each fold: 2, 4, 8, 16. Solve 2ⁿ = 16 and compare with your folds. Then explain why 2ⁿ = 0 has no solution.
Practice
1
Solve 5^(x−2) = 125.
125 = 5³, so x − 2 = 3 and x = 5. Check: 5³ = 125.
2
Solve 9^(x+2) = 3ˣ.
9 = 3², so 3^(2x+4) = 3ˣ, 2x + 4 = x and x = −4. Check: 9⁻² = 1/81 = 3⁻⁴.
3
Solve 4ˣ = 1/64.
1/64 = 4⁻³, so x = −3. Check: 4⁻³ = 1/64.
4
Solve 8^(x+1) = 32^(x−1).
2^(3x+3) = 2^(5x−5), so 3x + 3 = 5x − 5, 8 = 2x and x = 4. Check: 8⁵ = 2¹⁵ = 32³.
5
Solve 7ˣ = 1, and say why 3ˣ = −9 has no solution.
1 = 7⁰, so x = 0. A positive base never gives a negative number, so 3ˣ = −9 has no solution.
