Logarithm Properties
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Direct answer
A logarithm is an index first and everything else afterwards: log_a x = y means a^y = x, valid only for base a > 0 with a ≠ 1 and argument x > 0. All working rules descend from index laws — log(mn) = log m + log n, log(m/n) = log m − log n, log(m^n) = n log m — while the change-of-base identity log_a b = log_c b / log_c a converts any base into base 10 or e. In JEE use, every logarithmic equation is two problems in one: solve the underlying index equation, then delete every answer that violates the strict positivity of each argument.
What you must remember
- Definition with teeth: a^(log_a x) = x for x > 0, and log_a b × log_b a = 1 — reciprocal bases multiply to one, an instant simplification in MCQs.
- Change of base: log_a b = ln b / ln a, and the chain identity log_a b × log_b c = log_a c underlies most base-flipping questions.
- Base between 0 and 1: log base 1/2 is a decreasing function, so log_(1/2) x > 2 means 0 < x < 1/4 — the inequality direction reverses.
- Characteristic and mantissa: log N = characteristic + mantissa with the mantissa in [0, 1); for N = 0.0034 the characteristic is −3, written with a bar over 3 in the Indian textbook convention.
- Digit counting: the number of digits in 2^100 is the characteristic of 100 log 2 = 30.10, plus one — giving 31 digits using log 2 = 0.3010.
- No splitting sums: log(m + n) ≠ log m + log n; the product rule applies to multiplication only, the single most punished error in the chapter.
- Domain rule: every argument must be strictly positive before any manipulation; log(x − 1) + log(x − 2) lives only on x > 2.
Digit-counting with logarithms
How many digits does 2^100 have? Take common logarithms: log(2^100) = 100 × 0.3010 = 30.10. Whenever log N = k + m with k an integer and m in [0, 1), the number satisfies 10^k ≤ N < 10^(k+1), so it carries k + 1 digits — here 31. The reasoning, not the memory, is what transfers: any comparison of magnitudes (how many digits, which is larger, 3^40 or 4^30) reduces to comparing characteristics after converting to a common base.
The constants doing the quiet work are log 2 = 0.3010, log 3 = 0.4771 and log 7 = 0.8451 — slide-rule era values that JEE still expects as working knowledge. A related discipline: solving log_2(x − 1) + log_2(x − 2) = 1 requires the domain x > 2 first; the index equation gives (x − 1)(x − 2) = 2, so x^2 − 3x = 0 and x = 3 or x = 0, and x = 0 dies on the domain check.
Where marks leak
JEE Main tests quick evaluation — collapse an expression using the three index laws, flip a base, compare two logs. JEE Advanced hides logarithms inside functions and derivatives: a composite like log(x^2 + 1) whose monotonicity or range is asked. Three losses dominate: extraneous roots kept because the domain was never checked; inequality direction forgotten when the base lies between 0 and 1; and mixing log base 10 with log base e inside one computation. The bar-characteristic notation for negative characteristics also appears in the odd assertion-reason question, so reading 3̄ as −3, not 3, is a convention worth over-learning.
Frequently asked questions
What is log_a b times log_b a?
Exactly 1, since the two factors are reciprocals by the change-of-base rule.
Can log(m + n) be split into log m + log n?
No — the product rule applies to multiplication; sums inside a logarithm have no expansion.
How many digits does 2^100 contain?
31, because 100 log 2 = 30.10 and the characteristic 30 means k + 1 digits.
What happens to an inequality when the base is 1/2?
The direction reverses on removing the logarithm, since log base 1/2 is strictly decreasing.
What is the domain of log(x − 1) + log(x − 2)?
x > 2, because both arguments must be strictly positive simultaneously.