IB · Unit 1 · 1A Exponents
Rewrite both sides with the same base, then set the exponents equal.
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Same base
If \(a^m = a^n\), then the exponents are equal: \(m = n\).
\(2^x = 2^4\)
Bases are equal, so set exponents equal.
x = 4
A constant plus an exponential
\(5 + 2 \cdot 3^x = 23\)
Subtract 5 from both sides.
\(2 \cdot 3^x = 18\)
Divide both sides by 2.
\(3^x = 9\)
Change 9 to a power of 3.
\(3^x = 3^2\)
Bases are equal, so set exponents equal.
x = 2
One power times another
\(3^{x+4} = 9 \cdot 3^{2x}\)
Change 9 to a power of 3.
\(3^{x+4} = 3^2 \cdot 3^{2x}\)
Add the exponents, same as \(x^2 \cdot x^3 = x^5\).
\(3^{x+4} = 3^{2+2x}\)
The bases already match, or they are easy to rewrite.
\(2^x = 16\)
Change 16 to a power of 2.\(2^x = 2^4\)
Bases are equal, so set exponents equal.\(x = 4\)
Answer: x = 4
\(4^x = 64\)
Change 4 and 64 to powers of 2.\((2^2)^x = 2^6\)
Multiply the exponents.\(2^{2x} = 2^6\)
Bases are equal, so set exponents equal.\(2x = 6\)
Divide both sides by 2.\(x = 3\)
Answer: x = 3
\(5^x = 25\)
Change 25 to a power of 5.\(5^x = 5^2\)
Bases are equal, so set exponents equal.\(x = 2\)
Answer: x = 2
\(3^x = 9\)
Change 9 to a power of 3.\(3^x = 3^2\)
Bases are equal, so set exponents equal.\(x = 2\)
Answer: x = 2
\(2^x = 32\)
Change 32 to a power of 2.\(2^x = 2^5\)
Bases are equal, so set exponents equal.\(x = 5\)
Answer: x = 5
\(7^x = 7\)
7 is 7 to the first power.\(7^x = 7^1\)
Bases are equal, so set exponents equal.\(x = 1\)
Answer: x = 1
Rewrite both sides as the same base, then set the exponents equal.
\(2^{x+3} = 32\)
Change 32 to a power of 2.\(2^{x+3} = 2^5\)
Bases are equal, so set exponents equal.\(x+3 = 5\)
Subtract 3 from both sides.\(x = 2\)
Answer: x = 2
\(3^{x-2} = 81\)
Change 81 to a power of 3.\(3^{x-2} = 3^4\)
Bases are equal, so set exponents equal.\(x-2 = 4\)
Add 2 to both sides.\(x = 6\)
Answer: x = 6
\(5^{2x} = 125\)
Change 125 to a power of 5.\(5^{2x} = 5^3\)
Bases are equal, so set exponents equal.\(2x = 3\)
Divide both sides by 2.\(x = \dfrac{3}{2}\)
Answer: \(x = \dfrac{3}{2}\)
\(2^{x+1} = \dfrac{1}{8}\)
Change 1/8 to a power of 2.\(2^{x+1} = 2^{-3}\)
Bases are equal, so set exponents equal.\(x+1 = -3\)
Subtract 1 from both sides.\(x = −4\)
Answer: x = −4
\(3^{x+2} = \dfrac{1}{9}\)
Change 1/9 to a power of 3.\(3^{x+2} = 3^{-2}\)
Bases are equal, so set exponents equal.\(x+2 = -2\)
Subtract 2 from both sides.\(x = −4\)
Answer: x = −4
\(5^{x-1} = \dfrac{1}{125}\)
Change 1/125 to a power of 5.\(5^{x-1} = 5^{-3}\)
Bases are equal, so set exponents equal.\(x-1 = -3\)
Add 1 to both sides.\(x = −2\)
Answer: x = −2
\(4^{x-1} = 64\)
Change 4 and 64 to powers of 2.\((2^2)^{x-1} = 2^6\)
Multiply the exponents.\(2^{2x-2} = 2^6\)
Bases are equal, so set exponents equal.\(2x-2 = 6\)
Add 2 to both sides.\(2x = 8\)
Divide both sides by 2.\(x = 4\)
Answer: x = 4
\(9^x = 27\)
Change 9 and 27 to powers of 3.\((3^2)^x = 3^3\)
Multiply the exponents.\(3^{2x} = 3^3\)
Bases are equal, so set exponents equal.\(2x = 3\)
Divide both sides by 2.\(x = \dfrac{3}{2}\)
Answer: \(x = \dfrac{3}{2}\)
Different bases: rewrite both as a power of the same prime, then set the exponents equal.
\(4^x = 2^{x+4}\)
Change 4 to a power of 2.\((2^2)^x = 2^{x+4}\)
Multiply the exponents.\(2^{2x} = 2^{x+4}\)
Bases are equal, so set exponents equal.\(2x = x+4\)
Subtract x from both sides.\(2x - x = 4\)
\(x = 4\)
Answer: x = 4
\(4^x = 2^{x+6}\)
Change 4 to a power of 2.\((2^2)^x = 2^{x+6}\)
Multiply the exponents.\(2^{2x} = 2^{x+6}\)
Bases are equal, so set exponents equal.\(2x = x+6\)
Subtract x from both sides.\(2x - x = 6\)
\(x = 6\)
Answer: x = 6
\(9^x = 3^{x+4}\)
Change 9 to a power of 3.\((3^2)^x = 3^{x+4}\)
Multiply the exponents.\(3^{2x} = 3^{x+4}\)
Bases are equal, so set exponents equal.\(2x = x+4\)
Subtract x from both sides.\(2x - x = 4\)
\(x = 4\)
Answer: x = 4
\(25^x = 5^{x+3}\)
Change 25 to a power of 5.\((5^2)^x = 5^{x+3}\)
Multiply the exponents.\(5^{2x} = 5^{x+3}\)
Bases are equal, so set exponents equal.\(2x = x+3\)
Subtract x from both sides.\(2x - x = 3\)
\(x = 3\)
Answer: x = 3
\(8^x = 4^{x+1}\)
Change 8 and 4 to powers of 2.\((2^3)^x = (2^2)^{x+1}\)
Multiply the exponents.\(2^{3x} = 2^{2(x+1)}\)
Distribute: 2(x+1) = 2x+2.\(2^{3x} = 2^{2x+2}\)
Bases are equal, so set exponents equal.\(3x = 2x+2\)
Subtract 2x from both sides.\(3x-2x = 2\)
\(x = 2\)
Answer: x = 2
\(16^x = 8^{x+1}\)
Change 16 and 8 to powers of 2.\((2^4)^x = (2^3)^{x+1}\)
Multiply the exponents.\(2^{4x} = 2^{3(x+1)}\)
Distribute: 3(x+1) = 3x+3.\(2^{4x} = 2^{3x+3}\)
Bases are equal, so set exponents equal.\(4x = 3x+3\)
Subtract 3x from both sides.\(4x-3x = 3\)
\(x = 3\)
Answer: x = 3
\(27^x = 9^{x+1}\)
Change 27 and 9 to powers of 3.\((3^3)^x = (3^2)^{x+1}\)
Multiply the exponents.\(3^{3x} = 3^{2(x+1)}\)
Distribute: 2(x+1) = 2x+2.\(3^{3x} = 3^{2x+2}\)
Bases are equal, so set exponents equal.\(3x = 2x+2\)
Subtract 2x from both sides.\(3x-2x = 2\)
\(x = 2\)
Answer: x = 2
Move extra terms first, or rewrite a coefficient as a power of the same base.
\(6 + 3 \cdot 3^x = 33\)
Subtract 6 from both sides.\(3 \cdot 3^x = 27\)
Divide both sides by 3.\(3^x = 9\)
Change 9 to a power of 3.\(3^x = 3^2\)
Bases are equal, so set exponents equal.\(x = 2\)
Answer: x = 2
\(4 + 5 \cdot 2^x = 84\)
Subtract 4 from both sides.\(5 \cdot 2^x = 80\)
Divide both sides by 5.\(2^x = 16\)
Change 16 to a power of 2.\(2^x = 2^4\)
Bases are equal, so set exponents equal.\(x = 4\)
Answer: x = 4
\(2^{x+5} = 4 \cdot 2^{2x}\)
Change 4 to a power of 2.\(2^{x+5} = 2^2 \cdot 2^{2x}\)
Add the exponents, like x^2 · x^3 = x^5.\(2^{x+5} = 2^{2+2x}\)
Bases are equal, so set exponents equal.\(x+5 = 2+2x\)
Subtract 2 from both sides.\(x+3 = 2x\)
Subtract x from both sides.\(3 = x\)
\(x = 3\)
Answer: x = 3
\(2 \cdot 5^x + 10 = 60\)
Subtract 10 from both sides.\(2 \cdot 5^x = 50\)
Divide both sides by 2.\(5^x = 25\)
Change 25 to a power of 5.\(5^x = 5^2\)
Bases are equal, so set exponents equal.\(x = 2\)
Answer: x = 2
\(8^x = 2 \cdot 4^{2x}\)
Change 8 and 4 to powers of 2.\((2^3)^x = 2^1 \cdot (2^2)^{2x}\)
Multiply the exponents, like (x^2)^3 = x^6.\(2^{3x} = 2 \cdot 2^{4x}\)
Add the exponents, like x^2 · x^3 = x^5.\(2^{3x} = 2^{1+4x}\)
Bases are equal, so set exponents equal.\(3x = 1+4x\)
Subtract 4x from both sides.\(3x-4x = 1\)
\(-x = 1\)
Multiply both sides by −1.\(x = −1\)
Answer: x = −1
\(9^x = 27 \cdot 3^{x}\)
Change 9 and 27 to powers of 3.\((3^2)^x = 3^3 \cdot 3^{x}\)
Multiply the exponents, like (x^2)^3 = x^6.\(3^{2x} = 3^3 \cdot 3^{x}\)
Add the exponents, like x^2 · x^3 = x^5.\(3^{2x} = 3^{3+x}\)
Bases are equal, so set exponents equal.\(2x = 3+x\)
Subtract x from both sides.\(2x-x = 3\)
\(x = 3\)
Answer: x = 3
\(4^x = 8 \cdot 2^{3x}\)
Change 4 and 8 to powers of 2.\((2^2)^x = 2^3 \cdot 2^{3x}\)
Multiply the exponents, like (x^2)^3 = x^6.\(2^{2x} = 2^3 \cdot 2^{3x}\)
Add the exponents, like x^2 · x^3 = x^5.\(2^{2x} = 2^{3+3x}\)
Bases are equal, so set exponents equal.\(2x = 3+3x\)
Subtract 3x from both sides.\(2x-3x = 3\)
\(-x = 3\)
Multiply both sides by −1.\(x = −3\)
Answer: x = −3
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