Logarithms Quiz: Challenge Your Exponentiation Skills

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

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

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

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

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

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

Logarithms are the inverse operation of exponentiation. In other words, we need to find the exponent xxx such that:

10^x=100

We know that 10²=100, so the value of the logarithm is 2. Therefore, log₁₀ 100 = 2.

The logarithmic property of logₐ(xy) is based on the product rule, which states:

log⁡_a(xy)=log⁡_ax+log⁡_ay

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.

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:

10^x=0.01

We know that:

10⁻²=0.01

So, the value of log₁₀ 0.01 is -2. Therefore, the correct answer is -2.

To convert an exponential equation to logarithmic form, we use the following rule:

b^y=x can be written as log_⁡b 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:

log⁡₁₀1000=3

This means "10 raised to the power of 3 equals 1000."