1.
Log_{3}x=2, x = ?
Correct Answer
C. 9
Explanation
Logarithms are essentially the inverse of exponents. When you see log₃x = 2, it's asking the question: "To what power (exponent) must we raise the base (3) to get the result (x)?"
Converting to Exponential Form: To solve, it's often helpful to rewrite the logarithmic equation in its equivalent exponential form. The general pattern is:
logₐb = c <=> aᶜ = b
Applying this to our equation:
log₃x = 2 <=> 3² = x
Solving: Now it's a simple calculation: 3² = 9, so x = 9.
Answer: 9
2.
Log_{x}32 =5, x=?
Correct Answer
D. 2
Explanation
This equation is a bit different because the unknown (x) is the base of the logarithm.
Converting to Exponential Form: Again, let's rewrite in exponential form:
logₓ32 = 5 <=> x⁵ = 32
Solving: To find x, we need to think: "What number, when raised to the power of 5, equals 32?" The answer is 2, since 2⁵ = 32.
Answer: 2
3.
Log_{x}125=3, x = ?
Correct Answer
B. 5
Explanation
Understanding: Similar to the previous question, the unknown is the base. We need to find the number that, when raised to the power of 3, equals 125.
Converting to Exponential Form:
logₓ125 = 3 <=> x³ = 125
Solving: The cube root of 125 is 5 (5 x 5 x 5 = 125), so x = 5.
Answer: 5
4.
Log_{5}x = 3, x = ?
Correct Answer
B. 125
Explanation
This is of the more common form where the unknown is the result. What do we get when we raise 5 to the power of 3?
Converting to Exponential Form:
log₅x = 3 <=> 5³ = x
Solving: 5³ = 5 x 5 x 5 = 125, so x = 125.
Answer: 125
5.
Log_{y}512 = 3, y = ?
Correct Answer
A. 8
Explanation
The unknown is the base. We need to find the number that, when raised to the power of 3, results in 512.
Converting to Exponential Form:
logᵧ512 = 3 <=> y³ = 512
Solving: The cube root of 512 is 8 (8 x 8 x 8 = 512), so y = 8.
Answer: 8
6.
What is the value of log_{10}100?
Correct Answer
C. 2
Explanation
Logarithms are the inverse operation of exponentiation. In other words, we need to find the exponent xxx such that:
10x=100
We know that 102=100, so the value of the logarithm is 2. Therefore, log₁₀ 100 = 2.
7.
Which of the following is the logarithmic form of 1000 = 10³?
Correct Answer
A. Log 1000 = 10³
Explanation
To convert an exponential equation to logarithmic form, we use the following rule:
by=x can be written as logb x=y
In the equation 1000=103, the base b is 10, the exponent y is 3, and the result x is 1000. Thus, the logarithmic form is:
log101000=3
This means "10 raised to the power of 3 equals 1000."
8.
If logₐ x = 3, what is the value of x?
Correct Answer
A. X = a³
Explanation
Logarithms are the inverse of exponentiation.
Solving for x
To find the value of x, we rewrite the logarithmic equation in exponential form:
a³ = x
Therefore, x = a³
9.
Which of the following is the logarithmic property of logₐ(xy)?
Correct Answer
A. Logₐ x + logₐ y
Explanation
The logarithmic property of logₐ(xy) is based on the product rule, which states:
loga(xy)=logax+logay
This means that the logarithm of a product is equal to the sum of the logarithms of the factors. So if you have a product inside the logarithm (logₐ(xy)), you can break it down into the sum of two separate logarithms, logₐ x and logₐ y.
10.
What is the value of log₁₀ 0.01?
Correct Answer
A. -2
Explanation
The logarithmic expression log₁₀ 0.01 asks, "To what power must 10 be raised to result in 0.01?" In other words, we need to find the exponent x such that:
10x=0.01
We know that:
10−2=0.01
So, the value of log₁₀ 0.01 is -2. Therefore, the correct answer is -2.