Logarithms

For some positive real number $$ x $$, if $$ \log_{\sqrt{3}}(x) + \frac{\log_x(25)}{\log_x(0.008)} = \frac{16}{3}, $$...

If $$ X $$ and $$ Y $$ are positive real numbers such that $$ \log_X (X^2 + 12) = 4 $$ and $$ 3\log_Y X = 1 $$, then $$...

If $$\left(x + 6\sqrt{2}\right)^{\frac{1}{2}} - \left(x - 6\sqrt{2}\right)^{\frac{1}{2}} = 2\sqrt{2},$$ then $$x$$ e...

If $$ a, b $$ and $$ c $$ are positive real numbers such that $$ a > 10 \geq b \geq c $$ and $$ \frac{\log_8 (a + b)}{\...

The sum of all distinct real values of $$ x $$ that satisfy the equation $$ 10^x + \frac{4}{10^x} = \frac{81}{2}, $$ ...

For a real number $$ x $$, if $$ \frac{1}{2}, \frac{\log_4(x^2 - 9)}{\log_4 4}, \text{ and } \frac{\log_4\left(\frac{x^...

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

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

The number of distinct integer values of $$ n $$ satisfying $$ \frac{4 - \log_2 n}{3 - \log_4 n} < 0 $$ is ...