Logarithm Study Material for JEE Mains and Advance

Logarithms: JEE Mains & Advanced Study Material

1. Definition and Properties of Logarithm

The logarithm of a number x with respect to base b is the exponent to which b must be raised to yield x. Mathematically:

logbx = y ⇔ by = x

If \( a^x = b \), then \( x = \log_a b \). Here, a is the base, b is the argument, and x is the logarithm.

Properties:

  • \( \log_a 1 = 0 \)
  • \( \log_a a = 1 \)
  • \( \log_a a^x = x \)
  • \( a^{\log_a x} = x \)
  • logb(xy) = logbx + logby
  • logb(x/y) = logbx – logby
  • logb(xn) = n logbx
  • alogbc = clogba

Example 1:

Simplify \( \log_3 81 \).

Solution: \( \log_3 81 = \log_3 (3^4) = 4 \)

Example 2:

Find \( x \) if \( \log_2 x = 5 \).

Solution: \( x = 2^5 = 32 \)

Example 3:

Simplify: log28 + log24 – log216

Solution: log2(8×4) – log216 = log232 – log216 = log2(32/16) = log22 = 1

Practice Problems:

  1. \( \log_5 125 \)
  2. \( \log_{10} 0.01 \)
  3. \( \log_7 49 \)
  4. \( \log_2 0.25 \)
  5. \( \log_{16} 4 \)
  6. Simplify: log5125 + log525 – log5625 [Answer: 1]
  7. If log2 = a, express log216 in terms of a [Answer: 4a]
  8. Find the value of log10(0.001) [Answer: -3]
  9. Simplify: log381 – log39 [Answer: 2]
  10. If logba = 2 and logbc = 3, find logb(a2c3) [Answer: 13]

2. Fundamental Logarithmic Identities

Identities:

  • \( \log_a (mn) = \log_a m + \log_a n \)
  • \( \log_a \left(\frac{m}{n}\right) = \log_a m – \log_a n \)
  • \( \log_a (m^n) = n \log_a m \)
  • alogax = x
  • logaax = x
  • loga1 = 0
  • logaa = 1

Example 1:

Simplify \( \log_2 8 + \log_2 4 \).

Solution: \( \log_2 (8 \cdot 4) = \log_2 32 = 5 \)

Example 2:

Evaluate \( \log_3 81 – \log_3 9 \).

Solution: \( \log_3 \left(\frac{81}{9}\right) = \log_3 9 = 2 \)

Example 3:

Evaluate: 5log57 + 3log34

Solution: 5log57 = 7 and 3log34 = 4 ⇒ Total = 7 + 4 = 11

Practice Problems:

  1. \( \log_5 25 + \log_5 5 \)
  2. \( \log_2 64 – \log_2 8 \)
  3. \( 3\log_4 2 \)
  4. \( \log_6 36 – \log_6 6 \)
  5. \( 2\log_5 5 + \log_5 1 \)
  6. Evaluate: 10log10π [Answer: π]
  7. Simplify: 7log73 + 2log25 [Answer: 8]
  8. Find the value of eln 5 + 3log32 [Answer: 7]
  9. Show that 5log259 = 3
  10. Evaluate: (√2)log227 [Answer: 3√3]

3. Change of Base Formula

The change of base formula allows conversion between different logarithmic bases:

logba = logka / logkb

Mathematically: \( \log_a b = \frac{\log_c b}{\log_c a} \)

Example 1:

Evaluate \( \log_2 10 \) using base 10.

Solution: \( \frac{\log_{10} 10}{\log_{10} 2} = \frac{1}{0.3010} \approx 3.3219 \)

Example 2:

Evaluate log816 using base 2.

Solution: log816 = log216 / log28 = 4 / 3

Example 3:

If log32 = a, express log98 in terms of a.

Solution: log98 = log38 / log39 = 3log32 / 2 = 3a/2

Practice Problems:

  1. \( \log_4 8 \)
  2. \( \log_2 100 \)
  3. \( \log_5 25 \)
  4. \( \log_6 36 \)
  5. \( \log_3 81 \)
  6. Evaluate log279 using base 3 [Answer: 2/3]
  7. If log53 = a, express log2527 in terms of a [Answer: 3a/2]
  8. Find log45 × log56 × log67 × log78 [Answer: 3/2]
  9. Simplify: (log23)(log34)(log45)(log56)(log67)(log78) [Answer: 3]
  10. If log1227 = a, find log616 in terms of a [Answer: 4(3-a)/(3+a)]

4. Logarithmic Inequalities

Solve inequalities involving logarithms, ensuring domain restrictions are satisfied (argument > 0).

Example 1:

Solve \( \log_2 (x-1) > 3 \).

Solution: \( x – 1 > 2^3 \Rightarrow x > 9 \)

Example 2:

Solve \( \log_5 x < 2 \).

Solution: \( x < 25 \), and \( x > 0 \Rightarrow 0 < x < 25 \)

Practice Problems:

  1. \( \log_3 (x+2) > 1 \)
  2. \( \log_{10} x < 0 \)
  3. \( \log_2 (3x-1) \leq 4 \)
  4. \( \log_5 (x^2-1) > 0 \)
  5. \( \log_7 (x-3) \geq 1 \)

5. Characteristic and Mantissa

In \( \log_{10} N \), characteristic is the integer part, and mantissa is the decimal. For numbers > 1, characteristic = (number of digits before decimal – 1). For 0 < N < 1, characteristic = -(number of zeros after decimal + 1)

Example 1:

Find characteristic of \( \log_{10} 458 \).

Solution: Number of digits before decimal = 3, so characteristic = 2

Example 2:

Find characteristic of \( \log_{10} 0.00456 \).

Solution: 2 zeros after decimal → characteristic = -3

Practice Problems:

  1. Characteristic of \( \log_{10} 0.0089 \)
  2. Characteristic of \( \log_{10} 67.8 \)
  3. Characteristic of \( \log_{10} 0.00043 \)
  4. Characteristic of \( \log_{10} 789 \)
  5. Characteristic of \( \log_{10} 0.09 \)

6. Logarithmic Equations

Example 1:

Solve \( \log_2 (x + 1) = 3 \).

Solution: \( x + 1 = 8 \Rightarrow x = 7 \)

Example 2:

Solve \( \log_3 x + \log_3(x – 2) = 1 \).

Solution: \( \log_3[x(x-2)] = 1 \Rightarrow x^2 – 2x = 3 \Rightarrow x = 3 \) (x = -1 rejected)

Practice Problems:

  1. \( \log_5(x – 1) = 2 \)
  2. \( \log_2 x + \log_2 (x + 2) = 3 \)
  3. \( \log_4 (x^2 – 4) = 2 \)
  4. \( \log_3(x^2) = 4 \)
  5. \( \log_7 x = \log_7 9 \)

7. Applications in Calculus

Logarithms are used in differentiation and integration:

  • \( \frac{d}{dx}(\log x) = \frac{1}{x} \)
  • \( \int \log x \, dx = x \log x – x + C \)

Example 1:

Differentiate \( y = \log(x^2 + 1) \).

Solution: \( \frac{dy}{dx} = \frac{2x}{x^2 + 1} \)

Example 2:

Evaluate \( \int \log(2x) dx \).

Solution: Use integration by parts ⇒ \( x \log(2x) – x + C \)

Practice Problems:

  1. \( \frac{d}{dx}[\log(x^3)] \)
  2. \( \int \log(3x + 1) dx \)
  3. \( \int x \log x \, dx \)
  4. \( \frac{d}{dx}[\log(\sin x)] \)
  5. \( \frac{d}{dx}[\ln(e^x + 1)] \)

8. Logarithm with Complex Numbers

\( \log z = \ln|z| + i \arg z \) (principal branch)

Example 1:

Find \( \log(i) \).

Solution: \( |i| = 1 \), \( \arg(i) = \frac{\pi}{2} \Rightarrow \log(i) = 0 + i\frac{\pi}{2} \)

Example 2:

Find \( \log(-1) \).

Solution: \( \log(-1) = i\pi \)

Practice Problems:

  1. \( \log(1 + i) \)
  2. \( \log(-i) \)
  3. \( \log(-e^2) \)
  4. \( \log(\sqrt{3} + i) \)
  5. \( \log\left(\frac{1}{i}\right) \)

9. Advanced Problems for JEE Advanced

Example 1:

If \( \log_a b = 2 \) and \( \log_b c = 3 \), find \( \log_a c \).

Solution: \( \log_a c = \log_a b \cdot \log_b c = 2 \cdot 3 = 6 \)

Example 2:

Solve for x: xlogx(x2-x+1) > (x2-x+1)

Solution: Let y = x2-x+1. The inequality becomes xlogxy > y ⇒ No solution for x > 1, all x ∈ (0,1) are solutions.

Practice Problems:

  1. If \( \log_a b = x \) and \( \log_b a = y \), find the value of \( xy \).
  2. If \( \log_a b = m \), \( \log_b c = n \), express \( \log_a c \).
  3. Solve \( \log_3 x + \log_x 3 = 4 \)
  4. Solve: log2(9-2x) = 3 – x [Answer: x = 0, x = 3]
  5. Find all pairs (x,y) satisfying: logxy + logyx = 5/2 and xy = 64 [Answer: (2,32), (32,2), (4,16), (16,4)]