# Exponential and Logarithmic Equation

## Exponential Equation

Consider these equations $$4^x=\frac{1}{32}$$ $$3^{x-1}=2^x$$ $$2^{2x}-2^x-2=0$$ We call equations with variables in exponents exponential equation

We solve the above equations like this
$$4^x=\frac{1}{32}$$ $$2^{2x}=2^{-5}$$ $$2x = -5$$ $$x = -\frac{5}{2}$$

$$3^{x-1}=2^x$$ $$\log 3^{x-1}=\log 2^x$$ $$(x-1)\log 3=x\log 2$$ $$x\log 3 -\log 3=x\log 2$$ $$(\log 3-\log 2)x=\log 3$$ $$x=\frac{\log 3}{\log 3-\log 2}$$
$$2^{2x}-2^x-2=0$$
Let $$X=2^x$$
$$X^2-X-2=0$$ $$(X-2)(X+1) =0$$
Since $$X=2^x$$ can only be positive,
$$X=2$$ $$2^x=2$$
Therefore, $$x=1$$

### Practice

1. Solve for $$x$$ $$3^x2^{2-x}=6^x$$ 2. Solve for $$x$$ $$2^{x+1}\times 3=3^{2x+1}$$ 3. Solve for $$x$$ where $$x\gt 0$$ $$x^{\sqrt{x}}=\left(\sqrt{x}\right)^x$$ 4. Solve for $$x$$ where $$x\gt -1$$ $$(x+1)^x=3^x$$