What is a logarithm?

A logarithm is the power to which a base must be raised to yield a number. For example, log₃(27) = 3 because 3³ = 27.

What are the key properties of logarithms?

The key properties include: 1) Product Rule: log_b(m*n) = log_b(m) + log_b(n), 2) Quotient Rule: log_b(m/n) = log_b(m) - log_b(n), 3) Power Rule: log_b(m^k) = k * log_b(m).

How do you convert a logarithmic equation to exponential form?

To convert log_b(a) = c to exponential form, it becomes b^c = a. For instance, if log₅(25) = x, then it rewrites as 5^x = 25.

If log₂(x) = 5, what is x?

Hint: Convert to exponential form.

Convert to exponential form: 2^5 = x. Therefore, x = 32.

Solve for x: log₄(2x) = 3.

Hint: Convert to exponential form.

Convert to exponential form: 4^3 = 2x. Then 64 = 2x, so x = 32.

What is the natural logarithm of e?

Hint: Recall the definition of natural logarithm.

The natural logarithm of e is 1, since log_e(e) = 1.

If log₃(9) = x, what is x?

Hint: Express 9 as a power of 3.

Since 9 = 3², then log₃(9) = 2. Therefore, x = 2.

Evaluate: log₁₀(1000).

Hint: Write 1000 as a power of 10.

1000 = 10³, so log₁₀(1000) = 3.

If log₅(25) = a, what is log₅(5)?

Hint: Use known values.

Since 25 = 5², log₅(25) = 2. Therefore, log₅(5) = 1.

What is the change of base formula for logarithms?

The change of base formula states: log_b(a) = log_k(a) / log_k(b) for any positive k.

If log₁₀(2) = a, what is log₁₀(8)?

Hint: Express 8 in terms of powers of 2.

8 = 2³, so log₁₀(8) = 3 * log₁₀(2) = 3a.

If log₃(4) = a and log₃(2) = b, express log₃(8).

Hint: Use properties of logarithms.

Since 8 = 2³, log₃(8) = 3 * log₃(2) = 3b.

Solve for x: log₄(x) + log₄(4) = 3.

Hint: Use the product rule.

Using the product rule: log₄(4x) = 3. Thus, 4³ = 4x, leading to x = 16.

Evaluate log₈(64).

Hint: Write 64 as a power of 8.

Since 64 = 8², log₈(64) = 2.

What is the definition of a logarithm?

A logarithm is the power to which a base must be raised to obtain a given number. For instance, log_b(a) = c means b^c = a, where b is the base, a is the argument, and c is the logarithm.