If $\log_{10} 2 = 0.3010$, the value of $\log_{10} 80$ is

### Concept & Formula To find the logarithm of a composite number, you must break the number down into its prime factors, particularly those related to the given values and the logarithm's base. Here, we will decompose $80$ into factors of $2$ and $10$, and then apply the Product Rule and Power Rule of logarithms. Formulas used: 1. Product Rule: $\log (xy) = \log x + \log y$ 2. Power Rule: $\log (x^n) = n \log x$ ### Step-by-Step Solution Given the value: $$ \log_{10} 2 = 0.3010 $$ Step 1: Express $80$ using base $10$ and factors of $2$. Since $80 = 8 \times 10$, and $8 = 2^3$, we can write: $$ 80 = 2^3 \times 10 $$ Step 2: Apply the logarithm to both sides and use the Product Rule to separate the terms. $$ \log_{10} 80 = \log_{10} (2^3 \times 10) $$ $$ \log_{10} 80 = \log_{10} (2^3) + \log_{10} 10 $$ Step 3: Apply the Power Rule to the first term, bringing the exponent to the front. $$ \log_{10} 80 = 3 \log_{10} 2 + \log_{10} 10 $$ Step 4: Substitute the known values. We are given $\log_{10} 2 = 0.3010$, and we know intrinsically that $\log_{10} 10 = 1$. $$ \log_{10} 80 = 3(0.3010) + 1 $$ Step 5: Multiply and add to get the final result. $$ \log_{10} 80 = 0.9030 + 1 $$ $$ \log_{10} 80 = 1.9030 $$ ### Exam Strategy & Shortcut **Characteristic and Mantissa Shortcut:** Recognize that $80 = 8 \times 10^1$. The "$\times 10^1$" guarantees the answer will simply be the log of $8$ plus exactly $1$ (the characteristic). Mentally calculate $\log 8 = 3 \times \log 2 = 3 \times 0.3010 = 0.9030$. Add $1$ to the whole number place to account for the single zero in $80$, yielding $1.9030$ instantly. ### Common Pitfall A frequent trap is breaking $80$ down incorrectly or needlessly into smaller primes without utilizing the base ($10$). For instance, factoring $80$ as $2^4 \times 5$ requires you to calculate both $\log 2$ and $\log 5$, which introduces an extra step and a higher chance of arithmetic error compared to keeping the factor of $10$ intact. ### Final Answer **Therefore, the correct answer is 1.9030.**