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# an exponential equation that goes through (3,64) and (5,1024)

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an exponential equation that goes through (3,64) and (5,1024)

Guest Jan 21, 2015

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+20711
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an exponential equation that goes through (3,64) and (5,1024)

$$\small{\text{ Exponential equation is: \quad \boxed{y=y_0*a^x} }}\\ \small{\text{  (1): \quad 64 = y_0*a^3 \quad | \quad (3,64)  }}\\ \small{\text{  (2): \quad 1024 = y_0*a^5 \quad | \quad (5,1024)  }}\\\\ \begin{array}{lrcl} \hline \\ \dfrac{(2)}{(1)}: &\dfrac{1024}{64} &=& \dfrac{y_0a^5}{y_0a^3} \\\\ &\dfrac{1024}{64} &=& \dfrac{a^5}{a^3} \\ \\ &16 &=&a^{5-3} \\ &16 &=&a^{2} \quad | \quad \sqrt\\ &4 &=&a\\ &a &=&4 \end{array} \begin{array}{lrcl} \hline \\ (1): & 64 &=& y_0*4^3 \\ & 64 &=& y_0*64 \\ & 1 &=& y_0\\ & y_0 &=& 1\\ \end{array} \quad \small{\text{The exponential equation is: }}\boxed{y=1*4^x=4^x}$$$$$\\ \small{\text{ Proof: }}\\ \small{\text{ \quad y(3) = 4^3 = 64 \quad okay! }}\\ \small{\text{ \quad y(5) = 5^3 = 1024 \quad okay! }}$$ heureka Jan 21, 2015 #1 +20711 +10 Best Answer an exponential equation that goes through (3,64) and (5,1024) $$\small{\text{ Exponential equation is: \quad \boxed{y=y_0*a^x} }}\\ \small{\text{ (1): \quad 64 = y_0*a^3 \quad | \quad (3,64) }}\\ \small{\text{ (2): \quad 1024 = y_0*a^5 \quad | \quad (5,1024) }}\\\\ \begin{array}{lrcl} \hline \\ \dfrac{(2)}{(1)}: &\dfrac{1024}{64} &=& \dfrac{y_0a^5}{y_0a^3} \\\\ &\dfrac{1024}{64} &=& \dfrac{a^5}{a^3} \\ \\ &16 &=&a^{5-3} \\ &16 &=&a^{2} \quad | \quad \sqrt\\ &4 &=&a\\ &a &=&4 \end{array} \begin{array}{lrcl} \hline \\ (1): & 64 &=& y_0*4^3 \\ & 64 &=& y_0*64 \\ & 1 &=& y_0\\ & y_0 &=& 1\\ \end{array} \quad \small{\text{The exponential equation is: }}\boxed{y=1*4^x=4^x}$$$

$$\\ \small{\text{ Proof: }}\\ \small{\text{  \quad y(3) = 4^3 = 64 \quad okay!  }}\\ \small{\text{  \quad y(5) = 5^3 = 1024 \quad okay!  }}$$

heureka  Jan 21, 2015