To solve the inequality \( 12.37 < 1.6 \cdot 2^x \), we can follow these steps:
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Isolate the exponential term: Start by dividing both sides of the inequality by 1.6 (since 1.6 is positive, the direction of the inequality will not change):
\[ \frac{12.37}{1.6} < 2^x \]
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Calculate the left side:
\[ \frac{12.37}{1.6} \approx 7.73 \]
Therefore, the inequality simplifies to:
\[ 7.73 < 2^x \]
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Convert to logarithmic form: To solve for \( x \), we can take the logarithm (base 2) of both sides:
\[ x > \log_2(7.73) \]
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Calculate \( \log_2(7.73) \): Using the change of base formula \( \log_2(a) = \frac{\log_{10}(a)}{\log_{10}(2)} \) or \( \log_2(a) = \frac{\ln(a)}{\ln(2)} \):
\[ \log_2(7.73) \approx \frac{\log_{10}(7.73)}{\log_{10}(2)} \approx \frac{0.888}{0.301} \approx 2.95 \]
Alternatively, if using natural logarithms:
\[ \log_2(7.73) \approx \frac{\ln(7.73)}{\ln(2)} \approx \frac{2.048}{0.693} \approx 2.95 \]
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Conclusion:
The solution to the inequality \( 12.37 < 1.6 \cdot 2^x \) is:
\[ x > \log_2(7.73) \approx 2.95 \]
Thus, \( x \) must be greater than approximately 2.95.