Logarithmic Functions Practice Problems

Logarithmic functions are essential in mathematics for simplifying exponential equations. The basic logarithmic function is log_ex where e is the base of the logarithmic function. Logarithmic functions are the reverse function of exponentiation.

Solved Examples

These solved Logarithmic Functions Practice Problems will help you understand the concept and use them in a better manner.

Q1. Solve log₁₀ x = 3.

log₁₀ x = 3

Using the logarithm definition log_ax = p ⇔ x = a^p

x = 10³

x = 1000

Q2. Evaluate y = log₃5 + log₃4

y = log₃5 + log₃4

Using logarithmic function property

log(ab) = log a + log b

y = log₃(5 × 4)

y = log₃20

Q3. Solve log₅x + 20 = 22

log₅x + 20 = 22

log₅x = 22 - 20

log₅x = 2

Using property log_ax = p ⇔ x = a^p

x = 5²

x = 25

Q4. Solve z = log₇ 49 - log₇7

z = log₇ 49 - log₇7

Using property: log_a a = 1

z = log₇ 7² - 1

Using property: log a^b = b log a

z = 2 log₇ 7 - 1

Using property: log_a a = 1

z = 2 - 1

z = 1

Q5 . Evaluate p = log₆18 - log₆3

p = log₆18 - log₆3

Using property log(a/b) = log a - log b

p = log₆ (18 / 3)

p = log₆6

Using property: log_a a = 1

p = 1

Q6. Evaluate log₃(6x) = 2

log₃(6x) = 2

Using property log_a x = p ⇔ x = a^p

6x = 3²

6x = 9

x = 3/2

Q7. Solve log₄(16x - x²) = 3

log₄(16x - x²) = 3

Using property log_ax = p ⇔ x = a^p

(16x - x²) = 4³

(16x - x²) = 64

x² - 16x + 64 = 0

(x - 8)² = 0

x = 8

Q8. Solve c = log₉27.

c = log₉27

Using property: log_ba = log_xa / log_xb

c = log₃27 / log₃9

c = log₃3³ / log₃3²

c = (3log₃3) / (2log₃3)

Using property: log_a a = 1

c = 3/2

Q9. Find the value of x when log₂x + log₂(x + 6) = 4.

log₂x + log₂(x + 6) = 4

Using formula: log_b (pq) = log_b p + log_b q

log₂ [x (x + 6)] = 4

Using formula: a^x = p ⇔ x = log_ap

[x (x + 6)] = 2⁴

[x (x + 6)] = 16

x² + 6x – 16 = 0

x = 2 or – 8

Q10. Find the domain and range of given logarithmic function y = log (5x – 25) + 7.

y = log (5x – 25) + 7

To find the domain of the given function put 5x – 25 > 0

5x – 25 > 0

5x > 25

x > 5

Domain of the given logarithmic function = (5, ∞)

We know that,

Range of any logarithmic function is set of all real numbers.

Practice Problems

1. Solve: \log_{15} x = 4

2. Solve: y = \log_2 15 + \log_2 9

3. Solve: \log_2 x - 6 = 2

4. Solve: z = \log_8 128 - \log_8 8

5. Evaluate: p = \log_4 20 - \log_4 10

6. Solve: \log_5 (12x) = 2

7. Solve: \log_5 (10x - x^2) = 2

8. Solve: p = \log_8 16