1. **State the problem:** Solve the exponential equation algebraically. 2. **General approach:** For an equation of the form $$a^{x} = b^{x}$$ or $$a^{f(x)} = b^{g(x)}$$, we use properties of exponents and logarithms to isolate the variable. 3. **Example problem:** Suppose the equation is $$2^{x} = 8$$. 4. **Rewrite both sides with the same base:** Note that $$8 = 2^{3}$$, so the equation becomes $$2^{x} = 2^{3}$$. 5. **Set exponents equal:** Since the bases are equal and nonzero, the exponents must be equal: $$x = 3$$. 6. **If the equation is more complex, for example $$3^{2x+1} = 27$$, rewrite 27 as $$3^{3}$$: $$3^{2x+1} = 3^{3}$$. 7. **Set exponents equal:** $$2x + 1 = 3$$. 8. **Solve for $$x$$:** $$2x = 3 - 1$$ $$2x = 2$$ $$x = \frac{2}{2}$$ $$x = 1$$. 9. **If bases cannot be rewritten to be the same, use logarithms:** For example, solve $$5^{x} = 12$$. 10. **Take logarithm of both sides:** $$\log(5^{x}) = \log(12)$$. 11. **Use logarithm power rule:** $$x \log(5) = \log(12)$$. 12. **Isolate $$x$$:** $$x = \frac{\log(12)}{\log(5)}$$. This is the algebraic solution without a calculator. **Final answer depends on the specific equation given.** If you provide the exact equation, I can solve it step-by-step.