To solve the equation \( \log(105 \cdot 10^x) = 12 \), we can use the properties of logarithms.
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Start by applying the property of logarithms that allows us to separate the multiplication: \[ \log(105) + \log(10^x) = 12 \]
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The logarithm of \(10^x\) can be simplified as follows: \[ \log(10^x) = x \] Therefore, the equation now becomes: \[ \log(105) + x = 12 \]
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Next, we can isolate \(x\): \[ x = 12 - \log(105) \]
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Now, we need to calculate \(\log(105)\). Using a calculator: \[ \log(105) \approx 2.0212 \]
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Substitute this value back into the equation: \[ x \approx 12 - 2.0212 \approx 9.9788 \]
Finally, rounding to a reasonable number of decimal places gives: \[ x \approx 9.98 \]
Therefore, \(x \approx 9.98\).