Logarithmic equations with existence conditions

Statement

Solve the following equations after stating suitable existence conditions:

• (a) $\log_{2}(x)=5$
• (b) $\log_3(x^3)=-1$
• (c) $3+\log_{2}(x)=1-2\log_{2}(x)$
• (d) $-2+\dfrac{1}{2}\log\left(4-x^2\right)=0$

Solution

(a) Existence: $x>0$. $x=2^5=32$.

(b) Existence: $x>0$. $3\log_3 x=-1\Rightarrow \log_3 x=-\tfrac13\Rightarrow x=3^{-1/3}\approx 0.693$.

(c) Existence: $x>0$. $3\log_2 x=-2\Rightarrow \log_2 x=-\tfrac23\Rightarrow x=2^{-2/3}\approx 0.630$.

(d) Existence: $4-x^2>0\iff -2<x<2$. From the equation $\log(4-x^2)=4\Rightarrow 4-x^2=10^{4}=10000\Rightarrow x^2=-9996$: impossible. No real solution.

$\boxed{\text{(a) }x=32;\ \text{(b) }x=3^{-1/3};\ \text{(c) }x=2^{-2/3};\ \text{(d) no solution}}$