To four decimal places, $\log _{10} 5=0.6990$ and $\log _{10} 9=0.9542$. Evaluate the logarithm $\log _{10} \frac{5}{9}$ using these values. Do not use a calculator.
\[
\log _{10} \frac{5}{9}=\square
\]
(Simplify your answer.)

Answer

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Answer

Therefore, $\boxed{\log_{10} \left(\frac{5}{9}\right) = -0.2552}$

Steps

Step 1 ：Given: $\log_{10} 5 = 0.6990$ and $\log_{10} 9 = 0.9542$

Step 2 ：Using the property of logarithms, $\log_{10} \left(\frac{5}{9}\right) = \log_{10} 5 - \log_{10} 9$

Step 3 ：Substituting the given values, $\log_{10} \left(\frac{5}{9}\right) = 0.6990 - 0.9542$

Step 4 ：Simplifying the subtraction, $\log_{10} \left(\frac{5}{9}\right) = -0.2552$

Step 5 ：Therefore, $\boxed{\log_{10} \left(\frac{5}{9}\right) = -0.2552}$