If `log_(10)2=0.3010`, then the value of `log_(10)80` is

Options

A. A)`1.9030`
B. B)`1.6020`
C. C)`3.9030`
D. D)`2.9030`

Answer

To find the value of \( \log_{10} 80 \) given that \( \log_{10} 2 = 0.3010 \), we can follow these steps:

### Step 1: Rewrite \( \log_{10} 80 \)
We can express 80 as a product of its factors:
\[ 80 = 8 \times 10 \]
Thus, we can write:
\[ \log_{10} 80 = \log_{10} (8 \times 10) \]

### Step 2: Use the Product Rule of Logarithms
According to the product rule of logarithms, \( \log_b (xy) = \log_b x + \log_b y \):
\[ \log_{10} 80 = \log_{10} 8 + \log_{10} 10 \]

### Step 3: Simplify \( \log_{10} 10 \)
We know that \( \log_{10} 10 = 1 \):
\[ \log_{10} 80 = \log_{10} 8 + 1 \]

### Step 4: Rewrite \( \log_{10} 8 \)
Next, we can express 8 in terms of powers of 2:
\[ 8 = 2^3 \]
Thus, we can write:
\[ \log_{10} 8 = \log_{10} (2^3) \]

### Step 5: Use the Power Rule of Logarithms
According to the power rule of logarithms, \( \log_b (x^n) = n \cdot \log_b x \):
\[ \log_{10} 8 = 3 \cdot \log_{10} 2 \]

### Step 6: Substitute the Known Value
Now, we can substitute the value of \( \log_{10} 2 \):
\[ \log_{10} 8 = 3 \cdot 0.3010 = 0.9030 \]

### Step 7: Combine the Results
Now, we substitute this back into our equation for \( \log_{10} 80 \):
\[ \log_{10} 80 = 0.9030 + 1 = 1.9030 \]

### Final Answer
Thus, the value of \( \log_{10} 80 \) is:
\[ \log_{10} 80 = 1.9030 \]

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