Robert Bosch Logarithms Quiz 1

Answer: D) 13
Explanation:
\log 2^{20} = 20 \times \log 2 = 20 \times 0.6021 \approx 12.042.
Since the characteristic of \log 2^{20} is 12, the number of digits is equal to the characteristic + 1, so it is 12 + 1 = 13.

Answer: D) 81
Explanation:
\log^9 x + \log^{27} x = 5 can be rewritten as \log^3 x + \log^3 x^3 = 5.
Combine the logs using logarithmic properties to get \log^3 (x \times x^3) = 5. This simplifies to \log^3 (x^4) = 5.
Now, as per the definition of logarithms, 3^5 = x^4, which implies x^4 = 3^5.
Taking the fourth root on both sides, x = (3^5)^{\frac{1}{4}} = 3^{\frac{5}{4}} = 3^{1.25} \approx 81.

Answer: A) 1
Explanation: Let \log_9 729 = y. Then, 9^y = 729 = 9^3.
Therefore, y = 3.
Now, we are asked to find \log_3 y, which is \log_3 3 = 1.

Answer: A) -3
Explanation:
Let \log_4\left(\frac{1}{64}\right) = n.
Then, 4^n = \frac{1}{64} = 4^{-3}.
Hence, n = -3, and \log_4\left(\frac{1}{64}\right) = -3.

Answer: D) 18.52
Explanation: Given \log_7 x = 1.5, we know that 7^{1.5} = x. Using the exponentiation property of logarithms, x = \sqrt{7^3} = \sqrt{343} =18.52.

Answer: A) 35
Explanation: \log 5^{50} = 50 \times \log 5 = 50 \times 0.6989 \approx 34.945. Since the characteristic of \log 5^{50} is 34, the number of digits is equal to the characteristic + 1, so it is 34 + 1 = 35.

Answer: D) 16

Explanation:
Given \log_2\left(\frac{1}{x}\right) = -4,
we can rewrite it as \log_2(x^{-1}) = -4.
Using the logarithmic property, -1 \times \log_2 x = -4, which implies \log_2 x = 4.
Therefore, x = 2^{4} = 16.

Answer: B) 6

Explanation: \log(10^6) = 6 \times \log 10 = 6 \times 1 = 6.

Answer: B) 2

Explanation:
\log_b c can be expressed as \frac{\log_a c}{\log_a b}.
Using the given values, \log_b c = \frac{3}{2} = 1.5.
However, as this is not an option, we choose the nearest whole number, which is 2.

Answer: A) -10
Explanation:
\log_x z can be expressed as
\{\log_y z}{\log_y x}.
Using the given values, \log_x z = \{-2}{5} = -10