Logarithm Quiz: Ultimate Exam! MCQ Trivia

Explanation
The logarithms having base '10' are called common logarithms/briggesian logarithms. This is because the base 10 is commonly used in logarithmic calculations, especially in scientific and engineering fields. The common logarithm is denoted by log10 and is widely used for its convenience in calculations involving powers of 10. It is also called briggesian logarithm after the English mathematician Henry Briggs, who popularized the use of base 10 logarithms in the early 17th century.

Explanation
The logarithmic form of the equation 10^-3 = 0.001 is log base 10 of 0.001 equals -3, which can be written as log 0.001 = -3. This follows from the definition of a logarithm, which states that if 10^-3 = 0.001, then log base 10 of 0.001 is -3.

Explanation
The given relation y = logzx implies that raising z to the power of y will give us x.

Explanation
The given expression can be simplified using the properties of logarithms. The property states that when subtracting logarithms with the same base, we can divide the arguments of the logarithms. Applying this property, we can rewrite the expression as loga (x/y) + 3loga (z). This can be further simplified by using the property that states when multiplying a logarithm by a constant, we can raise the argument of the logarithm to that constant. Therefore, the expression can be expressed as loga (x/y) + loga (z^3). Combining the two logarithms, we get loga (xz^3/y), which matches the given answer.

Explanation
The given equation is in logarithmic form, where the base is 4 and the logarithm of 1/16 is equal to -2. To convert it into exponential form, we can rewrite it as 4^-2 = 1/16. This means that 4 raised to the power of -2 is equal to 1/16. Therefore, the answer is 4^-2 = 1/16.

Explanation
The given equation is asking us to find the value of y in the equation log9 (1/81) = y. In this equation, log9 represents the logarithm with base 9. The expression (1/81) is equal to 9 raised to the power of -2, which means that log9 (1/81) is equal to -2. Therefore, the unknown value y in the equation is -2.

Explanation
The equation log125x = 1/3 means that 125 raised to the power of something equals x. In this case, that something is 1/3. To solve for x, we need to find the value that, when raised to the power of 1/3, equals 125. The number that satisfies this condition is 5, because 5^3 = 125. Therefore, the answer is 5.

Explanation
The equation log9 (5x - 3) = log9 (3x + 7) implies that the logarithm of (5x - 3) to the base 9 is equal to the logarithm of (3x + 7) to the base 9. Since the bases are the same, the arguments must be equal. Therefore, 5x - 3 = 3x + 7. By solving this equation, we find that x = 5.

Explanation
The equation given is a logarithmic equation. To solve it, we can use the property of logarithms that states log(a) - log(b) = log(a/b). Applying this property to the equation, we get log((2 + x)/(x - 3)) = log(2). Since the logarithms are equal, the argument inside the logarithm should also be equal. Therefore, (2 + x)/(x - 3) = 2. By cross-multiplying and simplifying, we find that 2 + x = 2x - 6. Solving for x, we get x = 8. Therefore, the correct answer is 8.