Solve Exponential Equations

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

To solve $$2^x = 4^{x+1}$$ , rewrite 4 as $$2^2$$ : $$2^x = (2^2)^{x+1} = 2^{2(x+1)}$$ . Set the exponents equal: $$x = 2(x + 1)$$ , leading to $$x = -2$$ .

The exponent is 2, since $$8^2 = 64$$ .

Rewrite 27 as $$3^3$$ , so $$3^x = 3^3$$ . Therefore, $$x = 3$$ .

$$6^8$$ represents the product of 6 multiplied by itself 8 times: $$6 x 6 x 6 x 6 x 6 x 6 x 6 x 6$$ .

Since the bases are the same, set the exponents equal: $$3x + 3 = 6$$ . Solving gives $$x = 1$$ .

The base is 5.

The value of $$x$$ is 4, since $$2^4 = 16$$ .

Rewrite 4 as $$2^2$$ and 64 as $$2^6$$ : $$(2^2)^x = 2^6$$ , leading to $$2x = 6$$ , so $$x = 3$$ .

Rewrite 9 as $$3^2$$ : $$3^{2x} = 3^2$$ , so $$2x = 2$$ , leading to $$x = 1$$ .