Logarithmic Functions Practice Problems

Logarithmic functions are the reverse function of exponentiation. The basic logarithmic function is log_ex where e is the base of the logarithmic function. In this article, we will see the important formulas of logarithmic functions and solve some examples related to the logarithmic functions.

A logarithmic function is the inverse of an exponential function. It is typically written in the form y = log_⁡b(x), where b is the base of the logarithm, x is the argument, and y is the result. This means b^y = x.

Important Formulas on Logarithmic Functions

The table below represents the important formulas on Logarithmic Functions.

Example 1: Solve log₁₀ x = 3.

Solution:

log₁₀ x = 3

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

x = 10³

x = 1000

Example 2: Evaluate y = log₃5 + log₃4

Solution:

y = log₃5 + log₃4

Using logarithmic function property

log(ab) = log a + log b

y = log₃(5 × 4)

y = log₃20

Example 3: Solve log₅x + 20 = 22

Solution:

log₅x + 20 = 22

log₅x = 22 – 20

log₅x = 2

Using property log_ax = p ⇔ x = a^p

x = 5²

x = 25

Example 4: Solve z = log₇ 49 – log₇7

Solution:

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

Example 5: Evaluate p = log₆18 – log₆3

Solution:

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

Example 6: Evaluate log₃(6x) = 2

Solution:

log₃(6x) = 2

Using property log_ax = p ⇔ x = a^p

6x = 3²

6x = 9

x = 3/2

Example 7: Solve log₄(16x – x²) = 3

Solution:

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

Example 8: Solve c = log₉27.

Solution:

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

Example 9: Find the value of x when log₂x + log₂(x + 6) = 4.

Solution:

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

Example 10: Find the domain and range of given logarithmic function y = log (5x – 25) + 7.

Solution:

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.

Read More,

• Practice Questions on Data Handling
• Inverse Functions Practice Questions
• Functions Questions,

Q1: Solve log₁₅ x = 4.

Q2: Solve y = log₂15 + log₂9

Q3: Solve log₂x – 6 = 2

Q4: Solve z = log₈ 128 – log₈8

Q5: Evaluate p = log₄20 – log₄10

Q6: Evaluate log₅(12x) = 2

Q7: Solve log₅(10x – x²) = 2

Q8: Solve p = log₈(16)

What are Logarithmic Functions?

The reverse of exponentiation function is referred to as logarithmic function.

What is the Basic Logarithmic Function Example?

The basic logarithmic function example includes y = log₃ x.

What is Natural Logarithmic Function?

The logarithmic function with base e is called natural logarithmic function.

What are common bases used in logarithmic functions?

The most common bases used in logarithmic functions are:

• Base 10 (common logarithm): log⁡₁₀(x), often written as log⁡(x)
• Base e (natural logarithm): log⁡_e(x), often written as ln⁡(x).
• Base 2: log_⁡2(x)

What are the properties of logarithms?

Some important properties of logarithms include:

• Product Rule: log_⁡b(xy) = log_⁡b(x) + log⁡_b(y)
• Quotient Rule: log⁡_b(x/y)=log_⁡b(x) − log⁡_b(y)
• Power Rule: log⁡_b(x^k)=klog⁡_b(x)
• Change of Base Formula: log_⁡b(x)=log⁡_k(x)\log⁡_k(b)​, where k is any positive number

What is the domain and range of a logarithmic function?

• The domain of a logarithmic function y = log⁡_b(x) is x>0.
• The range of a logarithmic function y = log⁡_b(x) is all real numbers (−∞ < y < ∞)