Solving Exponential Equations

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

Convert 4 to a power of 2: $$2^{3n} = 2^2$$ . Therefore, $$3n = 2$$ , leading to $$n = \frac{2}{3}$$ .

Since the bases are the same, set the exponents equal: $$2x = -x$$ . Solving gives $$x = 0$$ .

First, rewrite 128 as $$2^7$$ . Then, express 4 as $$2^2$$ : $$2(2^2)^{2x+1} = 2^7$$ . This simplifies to $$2^{4x+3} = 2^7$$ , leading to $$4x + 3 = 7$$ , so $$x = 1$$ .

Since the bases are the same, set the exponents equal: $$x = 13$$ .

If $$a^m = a^n$$ , then $$m = n$$ , provided that @@a eq 0@@.

Identify if the bases can be made the same or if logarithms need to be used.

The logarithmic form is $$\text{log}_a(c) = b$$ .

To convert $$a^b = c$$ to logarithmic form, write it as $$\text{log}_a(c) = b$$ .

Rewrite 9 as $$3^2$$ . Therefore, $$x + 1 = 2$$ , leading to $$x = 1$$ .