If log ⁡ x + log ⁡ y = log ⁡ ( x + y ) \log x + \log y = \log (x + y) lo g x + lo g y = lo g ( x + y ) , find the relationship between x x x and y y y...

If $$ x $$ is a positive real number such that $$ 4\log_{10}x + 4\log_{100}x + 8\log_{1000}x = 13 $$...

What is the value of $$[\log_{10}(5 \cdot \log_{10} 100)]^2$$? ...

If $$ \log_x (0.1) = -\frac{1}{3} $$, find the value of $$ x $$. ...

Determine the value of $$ \log_2 \log_2 \log_3 \log_3 27^3 $$. ...

If $$ \log 3 $$, $$ \log (3^x - 2) $$, and $$ \log (3^x + 4) $$ are in arithmetic progression, find...

If $$ 3^a = 4 $$, $$ 4^b = 5 $$, $$ 5^c = 6 $$, $$ 6^d = 7 $$, $$ 7^e = 8 $$, and $$ 8^f = 9 $$, th...