Missing exponent?

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$\frac{256}{2^S}=64$

How would you solve for S?

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5 Answers

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Hint: $256=2^8$, $64=2^6$, and remember the rule $2^x/2^y = 2^{x-y}$.

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Here is one solution to your problem. $$\frac{256}{2^s}=64$$ $$256 = 64\cdot2^s$$ Since $64 = 2^6$ and $256 = 2^8$ the equation becomes: $$2^8 = 2^6\cdot 2^s$$ Simplify using the law: $a^b \cdot a^c = a^{b+c}$. $$2^8 = 2^{6+s}$$ Equate the exponents (this is only allowed when the bases are equal). $$8 = 6+s$$ $$2 = s$$ ...or equivalently: $$s = 2$$

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I would start by dividing by $64$ and multiplying by $2^S$ to give $2^S=\frac {256}{64}=4$. Converting $256$ and $64$ to powers of $2$ seems unnecessary complication to me.

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FURTHER HINT: $\mathrm{log}_2(2^{x}) = x$

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Well, think of $\frac{256}{2{^s}}$ as $256$ divided by $2^{s}$. then think of what multiplied by $64$ is equal to $256$. Now, $2$ to what power(s) will equal this number?

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