Exponential Equations

Solve for x over the real numbers:
3^(81 x) = 81^(3 x)

 

Take the natural logarithm of both sides and use the identity log(a^b) = b log(a):
81 log(3) x = 12 log(3) x

 

Subtract 12 x log(3) from both sides:
69 log(3) x = 0

 

Divide both sides by 69 log(3):
Answer: |x = 0

\(\begin{array}{rll}3^{81x}&=&81^{3x}\\ 81x\cdot \log_33 &=& 3x\cdot\log_381\\ 81x&=&12x\end{array}\\ \mbox{The only solution is }x=0\)

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