Exponential Equations (no logs)

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

Convert 64 to a power of 4: $$64 = 4^3$$ . Thus, $$p + 2 = 3$$ , leading to $$p = 1$$ .

The base is 8, which can be expressed as $$2^3$$ , and 4 can be expressed as $$2^2$$ .

$$25 = 5^2$$ .

Convert $$25$$ to $$5^2$$ : Thus, $$5^{(x+7)} = 5^{2x}$$ , leading to $$x + 7 = 2x$$ , so $$x = 7$$ .

Convert $$\frac{1}{125}$$ to $$5^{-3}$$ : Thus, $$5^{(2x+3)} = 5^{-3}$$ , leading to $$2x + 3 = -3$$ , so $$x = -3$$ .

Convert $$25$$ to $$5^2$$ : Thus, $$5^{-3x - 1} = 5^2$$ , leading to $$-3x - 1 = 2$$ , so $$x = -1$$ .

Identify if both sides can be expressed with the same base.

It means to manipulate the equation so that the variable is on one side and all other terms are on the other side.

Logarithms are the inverse operations of exponentiation. For example, if $$b^y = x$$ , then $$log_b(x) = y$$ .