Exponential Equations

Properties

Equal Bases:

am=an  ⟹  m=n;a>0∧a≠1 a^m = a^n \implies m = n; \quad a > 0 \land a \neq 1

Examples
  • 2x=25  ⟹  x=52^{x} = 2^{5} \implies x = 5
  • 32y=38  ⟹  2y=8  ⟹  y=43^{2y} = 3^{8} \implies 2y = 8 \implies y = 4
  • 103z−1=10z+7  ⟹  3z−1=z+7  ⟹  z=410^{3z - 1} = 10^{z + 7} \implies 3z - 1 = z + 7 \implies z = 4
  • (12)t=(12)4  ⟹  t=4\left(\frac{1}{2}\right)^{t} = \left(\frac{1}{2}\right)^{4} \implies t = 4
  • 7a+2=72a−3  ⟹  a+2=2a−3  ⟹  a=57^{a+2} = 7^{2a - 3} \implies a + 2 = 2a - 3 \implies a = 5
  • e3x=ex+10  ⟹  3x=x+10  ⟹  x=5e^{3x} = e^{x + 10} \implies 3x = x + 10 \implies x = 5

Equal Exponents:

am=bm  ⟹  a=b;m≠0,a>0∧b>0 a^m = b^m \implies a = b; \quad m \neq 0, a > 0 \land b > 0

Examples
  • x3=53  ⟹  x=5x^3 = 5^3 \implies x = 5
  • (2y)4=64  ⟹  2y=6  ⟹  y=3(2y)^4 = 6^4 \implies 2y = 6 \implies y = 3
  • a7=b7  ⟹  a=ba^7 = b^7 \implies a = b
  • (x+1)2=(3x−1)2(x+1)^2 = (3x - 1)^2 with x+1>0x+1 > 0, 3x−1>0  ⟹  x+1=3x−1  ⟹  x=13x - 1 > 0 \implies x+1 = 3x - 1 \implies x = 1
  • (4z)5=(2z+6)5  ⟹  4z=2z+6  ⟹  z=3(4z)^5 = (2z + 6)^5 \implies 4z = 2z + 6 \implies z = 3
  • (a2)6=(32)6  ⟹  a2=32  ⟹  a=3\left(\frac{a}{2}\right)^6 = \left(\frac{3}{2}\right)^6 \implies \frac{a}{2} = \frac{3}{2} \implies a = 3

Different Bases:

am=bn  ⟹  m=n=0 a^m = b^n \implies m = n = 0

Examples
  • 2x=7y2^x = 7^y and 2x=1  ⟹  x=02^x = 1 \implies x = 0, 7y=1  ⟹  y=07^y = 1 \implies y = 0
  • 3a+1=5b−23^{a+1} = 5^{b-2} and the value is 1 → a+1=0a+1 = 0, b−2=0  ⟹  a=−1b-2 = 0 \implies a = -1, b=2b = 2
  • 10m=9n=1  ⟹  m=010^m = 9^n = 1 \implies m = 0, n=0n = 0
  • (13)2x=4y+1=1  ⟹  2x=0\left(\frac{1}{3}\right)^{2x} = 4^{y+1} = 1 \implies 2x = 0, y+1=0  ⟹  x=0y+1 = 0 \implies x = 0, y=−1y = -1
  • πt=es=1  ⟹  t=0\pi^{t} = e^{s} = 1 \implies t = 0, s=0s = 0
  • 6x−3=82−y=1  ⟹  x−3=06^{x-3} = 8^{2-y} = 1 \implies x - 3 = 0, 2−y=0  ⟹  x=32 - y = 0 \implies x = 3, y=2y = 2

Explicit Equation by Reflexivity:

aa=bb  ⟹  a=b;a≠0∧b≠0a^a = b^b \implies a = b; \quad a \neq 0 \land b \neq 0

Examples
  • xx=22  ⟹  x=2x^x = 2^2 \implies x = 2
  • yy=33  ⟹  y=3y^y = 3^3 \implies y = 3
  • zz=11  ⟹  z=1z^z = 1^1 \implies z = 1
  • aa=(12)1/2  ⟹  a=12a^a = \left(\frac{1}{2}\right)^{1/2} \implies a = \frac{1}{2}
  • bb=44  ⟹  b=4b^b = 4^4 \implies b = 4
  • tt=55  ⟹  t=5t^t = 5^5 \implies t = 5

Explicit Equation by Symmetry:

aaa+m=aaa+n  ⟹  m=na^{a^{a+m}} = a^{a^{a+n}} \implies m = n

Examples
  • 222+m=222+3  ⟹  m=32^{2^{2+m}} = 2^{2^{2+3}} \implies m = 3
  • 333+x=333+1  ⟹  x=13^{3^{3+x}} = 3^{3^{3+1}} \implies x = 1
  • 222+y=222+y+0  ⟹  y=y2^{2^{2+y}} = 2^{2^{2+y+0}} \implies y = y (trivially valid: m=nm = n)
  • 444+a=444+0  ⟹  a=04^{4^{4+a}} = 4^{4^{4+0}} \implies a = 0
  • 555+t=555+2  ⟹  t=25^{5^{5+t}} = 5^{5^{5+2}} \implies t = 2
  • 101010+z=101010+7  ⟹  z=710^{10^{10+z}} = 10^{10^{10+7}} \implies z = 7

Special Case

xxm=m  ⟹  x=mm x^{x^m} = m \implies x = \sqrt[m]{m}

Equations with Variables in the Exponent

  1. Exponential Form:

xx=an  ⟹  x=a x^x = a^n \implies x = a

  1. Radical Form:

xx=n  ⟹  x=nx x^x = n \implies x = \sqrt[x]{n}