Evaluate 3/(0.534) using logarithms.

Step 1: Let the given expression be $P$.

$$P = \sqrt[3]{\frac{45.3 \times 0.00697}{0.534}}$$

We can write this as:

$$P = \left(\frac{45.3 \times 0.00697}{0.534}\right)^{\frac{1}{3}}$$

Step 2: Take the logarithm of both sides.

$$\log P = \log \left(\left(\frac{45.3 \times 0.00697}{0.534}\right)^{\frac{1}{3}}\right)$$

Using the logarithm property $\log(a^b) = b \log a$:

$$\log P = \frac{1}{3} \log \left(\frac{45.3 \times 0.00697}{0.534}\right)$$

Using the logarithm properties $\log\left(\frac{a}{b}\right) = \log a - \log b$ and $\log(ab) = \log a + \log b$:

$$\log P = \frac{1}{3} \left[ \log(45.3) + \log(0.00697) - \log(0.534) \right]$$

Step 3: Find the logarithms of each number using a 4-figure logarithm table.

• For $\log(45.3)$: Characteristic: There are 2 digits before the decimal point, so the characteristic is $2-1=1$. Mantissa: From log tables, the mantissa for 453 is 0.6561. So, $\log(45.3) = 1.6561$.

• For $\log(0.00697)$: Characteristic: The first non-zero digit (6) is in the third decimal place, so the characteristic is $\bar{3}$ (or -3). Mantissa: From log tables, the mantissa for 697 is 0.8432. So, $\log(0.00697) = \bar{3}.8432$.

• For $\log(0.534)$: Characteristic: The first non-zero digit (5) is in the first decimal place, so the characteristic is $\bar{1}$ (or -1). Mantissa: From log tables, the mantissa for 534 is 0.7275. So, $\log(0.534) = \bar{1}.7275$.

Step 4: Substitute these values into the equation for $\log P$ and perform the arithmetic.
First, calculate the sum of the logarithms in the numerator:

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