b) \( S_{1}\left(\frac{2}{3}\right)^{x-2}=\frac{8}{12 t} \) \( 6 .(0,25)^{x}=0,175 \) \[ \tau_{1}\left(O_{2}\right)^{x \rightarrow 0}=0,04 \] C \[ \begin{array}{l} 1,2^{x+1}+2^{x+2}=24 \\ 5,2 \cdot 8^{x+1}+3 \cdot 9^{x}=24 \end{array} \]

Solve the equation by following steps: - step0: Solve for \(x\): \(6\times 0.25^{x}=0.175\) - step1: Convert the expressions: \(6\left(\frac{1}{4}\right)^{x}=0.175\) - step2: Multiply by the reciprocal: \(6\left(\frac{1}{4}\right)^{x}\times \frac{1}{6}=0.175\times \frac{1}{6}\) - step3: Multiply: \(\left(\frac{1}{4}\right)^{x}=\frac{7}{240}\) - step4: Take the logarithm of both sides: \(\log_{\frac{1}{4}}{\left(\left(\frac{1}{4}\right)^{x}\right)}=\log_{\frac{1}{4}}{\left(\frac{7}{240}\right)}\) - step5: Evaluate the logarithm: \(x=\log_{\frac{1}{4}}{\left(\frac{7}{240}\right)}\) - step6: Simplify: \(x=\frac{-\log_{2}{\left(7\right)}+\log_{2}{\left(240\right)}}{2}\) - step7: Calculate: \(x=\frac{-\log_{2}{\left(7\right)}+4+\log_{2}{\left(15\right)}}{2}\) - step8: Simplify: \(x=\frac{\log_{2}{\left(\frac{15}{7}\right)}+4}{2}\) - step9: Evaluate: \(x=\frac{\log_{2}{\left(15\right)}-\log_{2}{\left(7\right)}+4}{2}\) Solve the equation \( S_{1}\left(\frac{2}{3}\right)^{x-2}=\frac{8}{12 t} \). Solve the equation by following steps: - step0: Solve for \(x\): \(S_{1}\left(\frac{2}{3}\right)^{x-2}=\frac{8}{12t}\) - step1: Calculate: \(S_{1}\left(\frac{2}{3}\right)^{x-2}=\frac{2}{3t}\) - step2: Multiply by the reciprocal: \(S_{1}\left(\frac{2}{3}\right)^{x-2}\times \frac{1}{S_{1}}=\frac{2}{3t}\times \frac{1}{S_{1}}\) - step3: Multiply: \(\left(\frac{2}{3}\right)^{x-2}=\frac{2}{3tS_{1}}\) - step4: Take the logarithm of both sides: \(\log_{\frac{2}{3}}{\left(\left(\frac{2}{3}\right)^{x-2}\right)}=\log_{\frac{2}{3}}{\left(\frac{2}{3tS_{1}}\right)}\) - step5: Evaluate the logarithm: \(x-2=\log_{\frac{2}{3}}{\left(\frac{2}{3tS_{1}}\right)}\) - step6: Move the constant to the right side: \(x=\log_{\frac{2}{3}}{\left(\frac{2}{3tS_{1}}\right)}+2\) - step7: Simplify: \(x=\log_{\frac{2}{3}}{\left(2\right)}-\log_{\frac{2}{3}}{\left(3\right)}-\log_{\frac{2}{3}}{\left(t\right)}-\log_{\frac{2}{3}}{\left(S_{1}\right)}+2\) Let's break down the problems step by step. ### Problem 1 **Equation:** \[ S_{1}\left(\frac{2}{3}\right)^{x-2}=\frac{8}{12 t} \] **Solution:** To solve for \( x \), we can rearrange the equation: \[ \left(\frac{2}{3}\right)^{x-2} = \frac{8}{12 t S_{1}} \] Taking logarithm on both sides: \[ x - 2 = \log_{\frac{2}{3}}\left(\frac{8}{12 t S_{1}}\right) \] Thus, \[ x = \log_{\frac{2}{3}}\left(\frac{8}{12 t S_{1}}\right) + 2 \] This can be expressed as: \[ x = \log_{\frac{2}{3}}{(2)} - \log_{\frac{2}{3}}{(3)} - \log_{\frac{2}{3}}{(t)} - \log_{\frac{2}{3}}{(S_{1})} + 2 \] ### Problem 2 **Equation:** \[ 6 \cdot (0.25)^{x} = 0.175 \] **Solution:** Rearranging gives: \[ (0.25)^{x} = \frac{0.175}{6} \] Calculating the right side: \[ (0.25)^{x} = \frac{0.175}{6} \approx 0.0291667 \] Taking logarithm: \[ x = \frac{\log_{2}{(15)} - \log_{2}{(7)} + 4}{2} \] ### Problem 3 **Equation:** \[ \tau_{1}\left(O_{2}\right)^{x \rightarrow 0}=0.04 \] **Solution:** Unfortunately, the service is currently unavailable for this calculation. ### Problem 4 **System of Equations:** \[ \begin{array}{l} 1.2^{x+1}+2^{x+2}=24 \\ 5.2 \cdot 8^{x+1}+3 \cdot 9^{x}=24 \end{array} \] **Solution:** This system of equations is also currently unsupported for calculation. ### Summary of Results 1. For the first equation, we have: \[ x = \log_{\frac{2}{3}}{(2)} - \log_{\frac{2}{3}}{(3)} - \log_{\frac{2}{3}}{(t)} - \log_{\frac{2}{3}}{(S_{1})} + 2 \] 2. For the second equation: \[ x = \frac{\log_{2}{(15)} - \log_{2}{(7)} + 4}{2} \] 3. The third equation could not be solved due to service unavailability. 4. The system of equations could not be solved due to unsupported calculation. If you have any specific values for \( S_{1} \), \( t \), or \( \tau_{1} \), please provide them for further assistance!