Question

In a tabular form, using logarithm table evaluate 15.47*0.085^2÷254^1÷3

Answer 1

To evaluate the expression \( 15.47 \times 0.085^2 \div 254^{1/3} \) using logarithm tables, we will first convert each part of the expression into its logarithmic form and then perform the operations.

### Step 1: Calculate \( 15.47 \)

Using the logarithm table, find \( \log_{10}(15.47) \).

### Step 2: Calculate \( 0.085^2 \)

Using the power rule for logarithms, \( \log_{10}(a^b) = b \cdot \log_{10}(a) \):
- First, find \( \log_{10}(0.085) \).
- Then calculate \( \log_{10}(0.085^2) = 2 \cdot \log_{10}(0.085) \).

### Step 3: Calculate \( 254^{1/3} \)

Using the power rule for logarithms again:
- Find \( \log_{10}(254) \).
- Then calculate \( \log_{10}(254^{1/3}) = \frac{1}{3} \cdot \log_{10}(254) \).

### Step 4: Combine using logarithmic rules

Now we'll combine all the logs as follows:
\[
\log_{10}(15.47 \times 0.085^2) - \log_{10}(254^{1/3})
\]

This can be expressed as:
\[
\log_{10}(15.47) + \log_{10}(0.085^2) - \log_{10}(254^{1/3})
\]
\[
= \log_{10}(15.47) + (2 \cdot \log_{10}(0.085)) - \frac{1}{3} \cdot \log_{10}(254)
\]

### Step 5: Calculate the values

Using a logarithm table or calculator, you would find these values:
- \( \log_{10}(15.47) \approx 1.188 \)
- \( \log_{10}(0.085) \approx -1.070 \)
- \( \log_{10}(254) \approx 2.404 \)

Now substituting these values:
\[
\log_{10}(15.47) \approx 1.188
\]
\[
\log_{10}(0.085^2) = 2 \cdot (-1.070) = -2.140
\]
\[
\log_{10}(254^{1/3}) = \frac{1}{3} \cdot 2.404 \approx 0.801
\]

### Step 6: Combine

Now add and subtract:
\[
1.188 + (-2.140) - 0.801
\]
\[
1.188 - 2.140 - 0.801 \approx -1.753
\]

### Step 7: Evaluate the final result

Now, to find the anti-logarithm:
\[
x = 10^{-1.753} \approx 0.0176
\]

### Final Result

| Operation | Value |
|----------------------------------|----------------------|
| \( \log_{10}(15.47) \) | \( 1.188 \) |
| \( \log_{10}(0.085) \) | \( -1.070 \) |
| \( \log_{10}(0.085^2) \) | \( -2.140 \) |
| \( \log_{10}(254) \) | \( 2.404 \) |
| \( \log_{10}(254^{1/3}) \) | \( 0.801 \) |
| Combined Logarithm | \( -1.753 \) |
| Final Result \( 15.47 \times 0.085^2 \div 254^{1/3} \) | \( \approx 0.0176 \) |

Therefore, the evaluated result is approximately \( 0.0176 \).