- Solve: \( \log_5 {x} + 3 \log_{25}{x} = 5 \)
Solution
\( \displaystyle 25 \)
- Solve: \( \log_2{(x + 2)} + 2\log_4{(x - 2)} = 5\)
Solution
\( \displaystyle 6 \)
- Simplify: \( \log_{10}{11} \cdot \log_{11}{12} \cdot \log_{12}{13} \cdot \ldots \cdot \log_{998}{999} \cdot \log_{999}{1000}\)
Solution
\( \displaystyle 3 \)
- Solve: \( \log_{3}{(x + 3)} + \log_{3}{(2x + 1)} \gt \log_{3}{12}\)
Solution
\( \displaystyle x \gt 1 \)
- Simplify: \( \frac{\log_{c}{a^{\log_{a}{c}}} - \log_{c}{c^2}}{\left( \log_{b^2}{d^2} \right) \left( \log_{d}{b^6} \right)} \)
Solution
\( \displaystyle -\frac{1}{6} \)
- Solve: \( \displaystyle e^{\ln{x^{\ln{4}}}} = 2\)
Solution
\( \displaystyle e^{\frac{2}{\ln{4}}} \)
- Solve: \( \log_{2}{(x^2 + 2x + 1)} \le 2^{\log_{5}{25}} \)
Solution
\( \displaystyle x \in [-5, -1) \cup (-1, 3] \)
- Simplify: \( \displaystyle \frac{e^{\ln{\left( \log_{7}{\left( \sqrt{2}^{\log_{2.5}{7}} \right)} \right)}}}{2\log_{4}{16} \cdot \frac{1}{\log_{3}{9}}} - \log_{\pi}{1} \)
Solution
\( \displaystyle \frac{1}{2} \)
- Solve: \( \displaystyle \log_{3}{(2x + 3) \ge \log_{9}{x^2}} + \log_{27}{x^3}\)
Solution
\( \displaystyle x \in (0, 3) \)
- Solve: \( \displaystyle \log_{4}{(x + 4) + \log_{16}{\left( x^2 - 8x + 16 \right)}} \le \log_{2}{\left( \frac{x^2 - 5x + 6}{x - 3} \right)} \)
Solution
\( \displaystyle x \le 5 \)
- Solve: \( \displaystyle \log_{7}{(x + 16)} + \log_{49}{\left( x^2 + 2x - 63\right)^2} \)
Solution
\( \displaystyle x = 8 \)
- Solve: \( \displaystyle \log_{2}{\left[ (2x - 6)(x + 2) \right]} - \log_{2}{(x^2 - x - 6)} \ge \log_{16}{(x + 6)} \)
Solution
\( \displaystyle x \in (-6, -2) \cup (3, 10] \)
- Solve: \( \displaystyle \log_{2}{\sqrt{\log_{2}{3}}} = \log_{4}{( \log_{4}{x}) } \)
Solution
\( \displaystyle x = 9 \)
- Solve: \( \displaystyle \log_{3}{\left( x\sqrt{27} \right)} = \left( \log_{3}{x} \right)^{-1}\)
Solution
\( \displaystyle x = \left\{ \sqrt{3}, \frac{1}{9} \right\} \)