Answer:
[tex]\boxed{\dfrac{\alpha\beta\gamma\theta}{\beta\gamma\theta+\alpha\gamma\theta+\alpha\beta\theta+\alpha\beta\gamma}}[/tex]
Step-by-step explanation:
Given log(x) to four different bases, a–d, you want the value of log(x) to the product of those bases.
Change of base
The formula for changing the base of a logarithm is ...
[tex]\log_a(x)=\dfrac{\log(x)}{\log(a)}\qquad\text{$=\alpha$ in this scenario}[/tex]
Application
Changing the base using the above formula, we have ...
[tex]\log_{abcd}(x)=\dfrac{\log(x)}{\log(abcd)}=\dfrac{\log(x)}{\log(a)+\log(b)+\log(c)+\log(d)}\\\\\\=\dfrac{1}{\dfrac{\log(a)}{\log(x)}+\dfrac{\log(b)}{\log(x)}+\dfrac{\log(c)}{\log(x)}+\dfrac{\log(d)}{\log(x)}}\\\\\\=\dfrac{1}{\dfrac{1}{\alpha}+\dfrac{1}{\beta}+\dfrac{1}{\gamma}+\dfrac{1}{\theta}}\\\\\\\boxed{\log_{abcd}(x)=\dfrac{\alpha\beta\gamma\theta}{\beta\gamma\theta+\alpha\gamma\theta+\alpha\beta\theta+\alpha\beta\gamma}}[/tex]
