\(log_ab=\dfrac{3}{2}\Rightarrow b=a^{\dfrac{3}{2}}=\left(\sqrt[]{a}\right)^3\)
Do b nguyên \(\Rightarrow\sqrt[]{a}\) nguyên \(\Rightarrow\sqrt[]{a}=x\Rightarrow a=x^2\)
\(log_cd=\dfrac{5}{4}\Rightarrow d=\left(\sqrt[4]{c}\right)^5\Rightarrow\sqrt[4]{c}\) nguyên \(\Rightarrow\sqrt[4]{c}=y\Rightarrow c=y^4\)
\(log_3\left(a-c\right)=2\Rightarrow a-c=9\)
\(\Rightarrow x^2-y^4=9\Rightarrow\left(x-y^2\right)\left(x+y^2\right)=9=1.9=9.1=3.3\)
Pt trên có cặp nghiệm nguyên dương duy nhất \(\left(x;y\right)=\left(5;2\right)\)
\(\Rightarrow log_3\left(b-d\right)=log_3\left(5^3-2^5\right)\)
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