Checkpoint: Solve with Common Logs Solution

Solve. Write the answer as a logarithm and as a number to four decimal places.

$300^{4 x} = 10$

$\begin{array}{rcl} \log 300^{4 x} & = & \log 10 \\ 4 x \log 300 & = & \log 10 \\ (\frac{1}{4}) 4 x & = & \frac{\log 10}{\log 300} (\frac{1}{4}) \\ x & = & \frac{\log 10}{4 \log 300} \approx 0.1009 \end{array}$

Note

Q: Will your answer be exactly 10 if you check with the decimal? Explain.

A: No, it will not be exactly 10 because the decimal is an approximate (rounded) value.

Q: Why is the approximate value of x smaller than one? Explain.

A: Since $300^{4} > 10$ , this means x needs to be much smaller than one to make them equal.