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 

title: Examples  
- $2^{x} = 2^{5} \implies x = 5$  
- $3^{2y} = 3^{8} \implies 2y = 8 \implies y = 4$  
- $10^{3z - 1} = 10^{z + 7} \implies 3z - 1 = z + 7 \implies z = 4$  
- $\left(\frac{1}{2}\right)^{t} = \left(\frac{1}{2}\right)^{4} \implies t = 4$  
- $7^{a+2} = 7^{2a - 3} \implies a + 2 = 2a - 3 \implies a = 5$  
- $e^{3x} = e^{x + 10} \implies 3x = x + 10 \implies x = 5$

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 

title: Examples  
- $x^3 = 5^3 \implies x = 5$  
- $(2y)^4 = 6^4 \implies 2y = 6 \implies y = 3$  
- $a^7 = b^7 \implies a = b$  
- $(x+1)^2 = (3x - 1)^2$ with $x+1 > 0$, $3x - 1 > 0 \implies x+1 = 3x - 1 \implies x = 1$  
- $(4z)^5 = (2z + 6)^5 \implies 4z = 2z + 6 \implies z = 3$  
- $\left(\frac{a}{2}\right)^6 = \left(\frac{3}{2}\right)^6 \implies \frac{a}{2} = \frac{3}{2} \implies a = 3$

Different Bases:

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

title: Examples  
- $2^x = 7^y$ and $2^x = 1 \implies x = 0$, $7^y = 1 \implies y = 0$  
- $3^{a+1} = 5^{b-2}$ and the value is 1 → $a+1 = 0$, $b-2 = 0 \implies a = -1$, $b = 2$  
- $10^m = 9^n = 1 \implies m = 0$, $n = 0$  
- $\left(\frac{1}{3}\right)^{2x} = 4^{y+1} = 1 \implies 2x = 0$, $y+1 = 0 \implies x = 0$, $y = -1$  
- $\pi^{t} = e^{s} = 1 \implies t = 0$, $s = 0$  
- $6^{x-3} = 8^{2-y} = 1 \implies x - 3 = 0$, $2 - y = 0 \implies x = 3$, $y = 2$

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 

title: Examples  
- $x^x = 2^2 \implies x = 2$  
- $y^y = 3^3 \implies y = 3$  
- $z^z = 1^1 \implies z = 1$  
- $a^a = \left(\frac{1}{2}\right)^{1/2} \implies a = \frac{1}{2}$  
- $b^b = 4^4 \implies b = 4$  
- $t^t = 5^5 \implies t = 5$

Explicit Equation by Symmetry:

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

title: Examples  
- $2^{2^{2+m}} = 2^{2^{2+3}} \implies m = 3$  
- $3^{3^{3+x}} = 3^{3^{3+1}} \implies x = 1$  
- $2^{2^{2+y}} = 2^{2^{2+y+0}} \implies y = y$ (trivially valid: $m = n$)  
- $4^{4^{4+a}} = 4^{4^{4+0}} \implies a = 0$  
- $5^{5^{5+t}} = 5^{5^{5+2}} \implies t = 2$  
- $10^{10^{10+z}} = 10^{10^{10+7}} \implies z = 7$

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}

equationzone.com