What is a Logarithm?
A logarithm is the inverse operation of exponentiation. It answers the fundamental question: "To what power must I raise base b to get x?"
The Exponential Scale Compressor
Logarithms act as a growth compressor. Instead of dealing with massive exponential jumps ($10^1 = 10, 10^6 = 1,000,000$), logarithms scale values down linearly (1, 2, 3, 4, 5, 6).
Logarithms appear in 1 to 2 direct questions in almost every CAT paper. Beyond direct questions, logarithm rules are essential for solving exponential series, compound interest growth rate doubling times, and digit calculation problems.
Connected Vault Modules:
Log of a product is the sum of individual logs.
Log of a quotient is the difference of individual logs.
Exponents on the argument bring down as multipliers.
Change base to any new base c, or invert argument and base.
Evaluating Simple Log Values
Q: Find the value of log_2(32) + log_10(0.001).
Step 1: Express 32 as power of base 2: 32 = 2^5 ⇒ log_2(2^5) = 5 · log_2(2) = 5(1) = 5.
Step 2: Express 0.001 as power of 10: 0.001 = 10^-3 ⇒ log_10(10^-3) = -3.
Step 3: Sum the values: 5 + (-3) = 2.
Base Change Telescoping Multiplication
Q: Evaluate the product log_3(5) × log_5(7) × log_7(81).
Step 1: Apply Base Change Theorem to common base 10:
(log 5 / log 3) × (log 7 / log 5) × (log 81 / log 7)
Step 2: Cancel telescoping terms log 5 and log 7:
= log 81 / log 3 = log_3(81)
Step 3: Since 81 = 3^4, log_3(3^4) = 4.
Solving Logarithmic Quadratic Equations
Q: Solve for real values of x: log_2(x) + log_2(x - 2) = 3.
Step 1 (Domain Constraint Check): Arguments must be positive: x > 0 and x - 2 > 0 ⇒ x > 2.
Step 2 (Combine Logs): log_2(x(x - 2)) = 3.
Step 3 (Exponential Form): x(x - 2) = 2^3 = 8 ⇒ x^2 - 2x - 8 = 0.
Step 4 (Factorize Quadratic): (x - 4)(x + 2) = 0 ⇒ x = 4 or x = -2.
Step 5 (Domain Verification): x = -2 violates x > 2. Therefore, the only valid root is x = 4.
❌ Trap 1: Log of Sum
log(x + y) ≠ log x + log y
Log of a sum DOES NOT split into a sum of logs. Addition of logs occurs ONLY when multiplying arguments inside: log(x · y) = log x + log y.
❌ Trap 2: Dividing Logs
(log x / log y) ≠ log(x - y)
Quotient of two logs is Base Change (log_y x). Subtracting logs is quotient of arguments log(x / y).
❌ Trap 3: Ignoring Domain
Forgetting x > 0 and b > 0, b ≠ 1
Always verify roots against domain constraints. Negative values created during algebraic steps must be discarded!
⚡ Power of Base & Argument Trick
Exponents on the argument go to the numerator; exponents on the base go to the denominator!
⚡ Digit Calculation Formula
To find how many digits 3^50 has: Calculate 50 × log_10(3) = 50 × 0.4771 = 23.855. Take floor (23) and add 1 = 24 Digits!
Question 1:
If log_3 x + log_9 x + log_27 x = 11, find the value of x.
Solution: Express bases in terms of power of 3:
log_3 x + log_{3^2} x + log_{3^3} x = 11
⇒ log_3 x + (1/2)log_3 x + (1/3)log_3 x = 11
⇒ log_3 x · (1 + 1/2 + 1/3) = 11 ⇒ log_3 x · (11 / 6) = 11
⇒ log_3 x = 6 ⇒ x = 3^6 = 729.
Question 2:
If log_10 2 = 0.3010, how many digits are there in 2^100?
Solution: Apply the Digit Formula: Digits = ⌊100 · log_10 2⌋ + 1.
= ⌊100 × 0.3010⌋ + 1 = ⌊30.10⌋ + 1 = 30 + 1 = 31 digits.
Question 3:
Solve for x: log_x(2x + 3) = 2.
Solution: Convert to exponential form: x^2 = 2x + 3.
x^2 - 2x - 3 = 0 ⇒ (x - 3)(x + 1) = 0 ⇒ x = 3 or x = -1.
Domain Check: Base of log must be positive (x > 0 and x ≠ 1). Discard x = -1. Therefore, x = 3.
Q: What is the fundamental definition of a logarithm?
A logarithm is the inverse operation of exponentiation. If b^y = x (where b > 0, b ≠ 1, x > 0), then log_b(x) = y. It answers what power base b must be raised to in order to produce x.
Q: What is the Base Change Theorem in Logarithms?
The Base Change Theorem states that log_b(a) = log_c(a) / log_c(b) for any valid positive base c. A useful corollary is log_b(a) = 1 / log_a(b).
Q: Why can't you take the logarithm of a negative number or zero?
Since positive bases raised to any real power always produce positive values (b^y > 0), log_b(x) is defined only for x > 0. The logarithm of zero or a negative number is undefined in real numbers.
Q: How do you calculate the number of digits in a large power like 3^50?
The number of digits in a^b is given by ⌊b · log_10(a)⌋ + 1. For 3^50, calculate 50 × log_10(3) = 50 × 0.4771 = 23.855. Taking floor(23.855) + 1 gives 23 + 1 = 24 digits.