7.5A Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent. For example, $$2^x = 8$$ .

Convert to the same base: $$3^{4x} = (3^2)^{x-7}$$ , which simplifies to $$3^{4x} = 3^{2(x-7)}$$ . Set the exponents equal: $$4x = 2(x-7)$$ . Solve for x to get $$x = -7$$ .

Add 3 to both sides to isolate the exponential term: $$-6e^{8x+8} = -20$$ .

Divide both sides by -6: $$e^{8x+8} = \frac{20}{6} = \frac{10}{3}$$ .

Use the property $$\text{if } e^y = a, \text{ then } y = \text{ln}(a)$$ . Thus, $$8x + 8 = \text{ln}\frac{10}{3}$$ .

Subtract 8 from both sides: $$8x = \text{ln}\frac{10}{3} - 8$$ , then divide by 8: $$x = \frac{\text{ln}\frac{10}{3} - 8}{8}$$ .

Add 6 to both sides: $$-10e^{2-2x} = -60$$ , then divide by -10: $$e^{2-2x} = 6$$ .

Take the natural logarithm: $$2-2x = \text{ln}(6)$$ , then isolate x: $$x = 1 - \frac{\text{ln}(6)}{2}$$ .

Convert 256 to base 4: $$256 = 4^4$$ , so the equation becomes $$4^{3x-2} = (4^4)^{2x}$$ .

Set the exponents equal: $$3x - 2 = 8x$$ , then solve for x: $$x = -\frac{2}{5}$$ .