Evaluate sqrt((3.894 × 0.0645)/(32.52)) using logarithm tables.

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: Evaluate using logarithm tables
Let $Y = \sqrt{\frac{3.894 \times 0.0645}{32.52}}$ .
Taking the logarithm base 10 on both sides:

\log Y = \log \left( \left(\frac{3.894 \times 0.0645}{32.52}\right)^{1/2} \right)

\log Y = \frac{1}{2} \log \left(\frac{3.894 \times 0.0645}{32.52}\right)

Using logarithm properties, $\log(A/B) = \log A - \log B$ and $\log(A \times B) = \log A + \log B$ :

\log Y = \frac{1}{2} [\log(3.894) + \log(0.0645) - \log(32.52)]

Find the logarithms of the individual numbers.
Using logarithm tables:
- $\log(3.894) = 0.5904$
- $\log(0.0645) = \bar{2}.8096$ (which is $-2 + 0.8096$ )
- $\log(32.52) = 1.5121$
Substitute these values into the equation for $\log Y$ .
$\log Y = \frac{1}{2} [0.5904 + (\bar{2}.8096) - 1.5121]$
First, calculate the sum of the logarithms in the numerator:
$\log(3.894 \times 0.0645) = 0.5904 + \bar{2}.8096 = 0.5904 + (-2 + 0.8096) = (0.5904 + 0.8096) - 2 = 1.4000 - 2 = \bar{1}.4000$
Next, subtract the logarithm of the denominator:
$\log\left(\frac{3.894 \times 0.0645}{32.52}\right) = \bar{1}.4000 - 1.5121$ $= (-1 + 0.4000) - 1.5121 = -1 - 1.5121 + 0.4000 = -2.5121 + 0.4000 = -2.1121$
Multiply by $\frac{1}{2}$ .
$\log Y = \frac{1}{2} (-2.1121) = -1.05605$
Convert $\log Y$ to characteristic-mantissa form (where the mantissa is positive).
$-1.05605 = -2 + (2 - 1.05605) = -2 + 0.94395$
So, $\log Y = \bar{2}.94395$ .
Find the antilog of $\bar{2}.94395$ .
The mantissa is 0.94395. From antilog tables, the number corresponding to 0.94395 is approximately 8.789.
The characteristic is $\bar{2}$ , which means the decimal point is two places to the left of the first digit.
$Y = 0.08789$
The value is $\boxed{0.08789}$ .

: Find the value of $x$ in the equation below.
$125^x \times 5^{3x} - 3125 = 0$
Isolate the terms with $x$ .
$125^x \times 5^{3x} = 3125$
Express all numbers as powers of the same base.
In this case, base 5.
We know that $125 = 5^3$ and $3125 = 5^5$ .
Substitute these into the equation:
$(5^3)^x \times 5^{3x} = 5^5$
Apply the exponent rules $(a^m)^n = a^{mn}$ and $a^m \times a^n = a^{m+n}$ .
$5^{3x} \times 5^{3x} = 5^5$ $5^{3x + 3x} = 5^5$ $5^{6x} = 5^5$
Equate the exponents since the bases are the same.
$6x = 5$
Solve for $x$ .
$x = \frac{5}{6}$
The value of $x$ is $\boxed{\frac{5}{6}}$ .

Evaluate sqrt((3.894 x 0.0645)/(32.52)) using logarithm tables.

Step-by-step solution

Find the logarithms of the individual numbers.

Substitute these values into the equation for $\log Y$ .

Multiply by $\frac{1}{2}$ .

Convert $\log Y$ to characteristic-mantissa form (where the mantissa is positive).

Find the antilog of $\bar{2}.94395$ .

Isolate the terms with $x$ .

Express all numbers as powers of the same base.

Apply the exponent rules $(a^m)^n = a^{mn}$ and $a^m \times a^n = a^{m+n}$ .

Equate the exponents since the bases are the same.

Solve for $x$ .

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More Mathematics Questions

Evaluate sqrt((3.894 x 0.0645)/(32.52)) using logarithm tables.

Step-by-step solution

Find the logarithms of the individual numbers.

Substitute these values into the equation for log⁡Y\log YlogY.

Multiply by 12\frac{1}{2}21​.

Convert log⁡Y\log YlogY to characteristic-mantissa form (where the mantissa is positive).

Find the antilog of 2ˉ.94395\bar{2}.943952ˉ.94395.

Isolate the terms with xxx.