1. **Problem Statement:** Solve the equation involving exponents and logarithms: $$2^{x} = 8$$ and find $x$. 2. **Formula and Rules:** Recall that $a^{m} = a^{n}$ implies $m = n$ if $a > 0$ and $a \neq 1$. Also, logarithms are the inverse of exponents: $$\log_{a}(a^{x}) = x$$. 3. **Step-by-step Solution:** - Express 8 as a power of 2: $$8 = 2^{3}$$. - So the equation becomes $$2^{x} = 2^{3}$$. - Since the bases are equal and positive, set the exponents equal: $$x = 3$$. 4. **Using Logarithms:** Alternatively, take the logarithm base 2 of both sides: $$\log_{2}(2^{x}) = \log_{2}(8)$$ Using the logarithm power rule: $$x \cdot \log_{2}(2) = \log_{2}(8)$$ Since $$\log_{2}(2) = 1$$, this simplifies to: $$x = \log_{2}(8)$$ Calculate $$\log_{2}(8)$$: Since $$8 = 2^{3}$$, $$\log_{2}(8) = 3$$. 5. **Final Answer:** $$x = 3$$. This example shows how to solve exponential equations by rewriting bases or using logarithms.