Questions: Logarithms Introduction

Question 1

What is log_2(32)?

• A. 4
• B. 5
• C. 16
• D. 6

Answer: (B)

log_2(32) asks: 'To what power must we raise 2 to get 32?' Since 2^5 = 32, the answer is 5. A common error is confusing log_2(32) with 32/2 = 16 or with 2 × 32. Logarithms are exponents, not multiplication or division.

Question 2

log(a + b) = log(a) + log(b) for any positive values of a and b.

• T. True
• F. False

Answer: False

This is one of the most common logarithm errors. The product rule states log(a × b) = log(a) + log(b) — multiplication inside the log becomes addition outside. There is no corresponding rule for log(a + b). For example, log(1 + 9) = log(10) = 1, but log(1) + log(9) = 0 + log(9) ≈ 0.954, which is different.

Question 3

Why is log_b(x) undefined when x = 0 or when x is negative?

Model answer: log_b(x) = y means b^y = x. For any positive base b, b^y is always positive regardless of what y is — you cannot raise a positive number to any real power and get 0 or a negative result. So there is no real exponent y that satisfies b^y = 0 or b^y = -5, making the logarithm undefined for non-positive inputs.

This connects directly to the prerequisite topic on exponential functions: the range of b^y (for b > 0, b ≠ 1) is the set of all positive real numbers. Since the logarithm is the inverse of the exponential, its domain must be exactly the range of the exponential — positive reals only. The domain restriction is a consequence of the inverse relationship, not an arbitrary rule.