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 \(x\gt -1\) $$ (x+1)^x=3^x $$