Aspect	Logarithmic Equations	Exponential Equations
Definition	Involves logarithms of variables	Involves variables as exponents
General Form	logb​(x) = y	bx = y
Base	The base b is the number being raised to a power	The base b is the number being raised to the exponent
Variable Location	Variable is inside the logarithm	Variable is in the exponent
Conversion	Convert to exponential form to solve	Convert to logarithmic form to solve
Solving Methods	Use properties of logarithms, convert to exponential	Use properties of exponents, convert to logarithmic
Graph Characteristics	Inverse of exponential functions, passes through (1, 0)	Rapid growth or decay, passes through (0, 1)
Example	log2​(x)=3	2x = 8
Application	Used in solving for exponents, growth rates, scales	Used in modeling growth, decay, and compound interest
Inverse Relationship	Logarithmic functions are the inverses of exponential functions	Exponential functions are the inverses of logarithmic functions

Applications of Logarithmic Equations

Logarithmic equations have a wide range of applications across various fields. Here are some key applications:

• pH Calculation: The pH of a solution is calculated using the logarithmic equation:

[Tex]\text{pH} = -\log_{10}[\text{H}^+][/Tex]

It is the concentration of hydrogen ions in the solution.

• Richter Scale: The Richter scale measures the magnitude of earthquakes using the logarithmic equation:

[Tex]M = \log_{10}\left(\frac{A}{A_0}\right) [/Tex]

Where M is the magnitude, A is the amplitude of the seismic waves, and A₀​ is a reference amplitude.

• Population Growth: Logarithmic equations model population growth, where the growth rate decreases as the population increases:

[Tex]N(t) = N_0 e^{rt}[/Tex]

Where N(t) is the population at time t, N₀​ is the initial population, and r is the growth rate.

• Sound Intensity: The decibel scale measures sound intensity using the logarithmic equation:

[Tex]L = 10 \log_{10} \left(\frac{I}{I_0}\right)[/Tex]

Where L is the sound level in decibels, I is the intensity of the sound, and I₀​ is the reference intensity.

Read More,

• Exponential Function
• Logarithmic Formulas
• Log Rules

Solved Examples of Logarithmic Equations

Example 1: Solve log₂ (x) = 3

Solution:

Convert the logarithmic equation to its exponential form:

2³ = x

⇒ 8 = x

Therefore, the solution is: x = 8

Example 2: Solve log₃ (x) + log₃ (x – 2) = 1.

Solution:

Use the product property of logarithms: log_b(mn) = log_b(m) + log_b(n)

log₃(x(x – 2)) = 1

Convert the logarithmic equation to its exponential form:

3¹ = x(x – 2)

Simplify and solve the quadratic equation:

3 = x² – 2x

⇒ x² – 2x – 3 = 0

Factor the quadratic equation:

(x – 3)(x + 1) = 0

Solve for x:

x = 3 or x = -1

Since the logarithm of a negative number is undefined, the solution is:

x = 3

Example 3: Solve log₅(x + 1) + log₅(x – 1) = 1.

Solution:

Use the product property of logarithms:

log₅ ((x + 1)(x – 1)) = 1

Convert the logarithmic equation to its exponential form:

5¹ = (x + 1)(x – 1)

Simplify the right-hand side:

5 = x² – 1

Solve the quadratic equation:

x² – 1 = 5

⇒ x² = 6

x = ± √6

Since log₅ (x + 1) and log₅ (x – 1) must both be defined, x must be greater than 1. Therefore, the solution is:

x = √6

Example 4: Solve log₂(x + 3) – log₂ (x – 1) = 2

Solution:

Use the quotient property of logarithms: log_b (m/n) = log_b (m) – log_b (n)

[Tex]\log_2 \left(\frac{x + 3}{x – 1}\right) = 2[/Tex]

Convert the logarithmic equation to its exponential form:

2² = [Tex] \frac{x + 3}{x – 1}[/Tex]

Simplify and solve the resulting equation:

4 = [Tex] \frac{x + 3}{x – 1}[/Tex]

⇒ 4(x – 1) = x + 3

⇒ 4x – 4 = x + 3

⇒ 3x = 7

⇒ x = 7/3

Example 5: Solve log₄ (2x – 1) + log₄ (x + 3) = 2

Solution:

Use the product property of logarithms:

log₄ ((2x – 1)(x + 3)) = 2

Convert the logarithmic equation to its exponential form:

4² = (2x – 1)(x + 3)

Simplify and solve the resulting quadratic equation:

16 = (2x – 1)(x + 3)

⇒ 16 = 2x² + 6x – x – 3

⇒ 16 = 2x² + 5x – 3

⇒ 2x² + 5x – 19 = 0

Solve the quadratic equation using the quadratic formula:

[Tex]x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}[/Tex]

Here, a = 2, b = 5, and c = -19:

[Tex]x = \frac{-5 \pm \sqrt{25 + 152}}{4}[/Tex]

⇒ [Tex]x = \frac{-5 \pm \sqrt{177}}{4}[/Tex]

Practice Problems on Logarithmic Equations

Problem 1: Solve for x: log⁡₃(x) = 4

Problem 2: Solve for x: log⁡₅(2x − 1) = 2

Problem 3: ⁡Solve for x: log₂(x + 3) + log⁡₂(x − 1) = 3

Problem 4: Solve for x: log⁡₄(x² − 2x) = 1

Problem 5: Solve for x: log⁡₇(x + 5) − log⁡₇(x − 1)=2

Problem 6: Solve for x: log⁡₆(3x + 4)=2

Problem 7: Solve for x: log⁡₂(x) + log⁡₂(x − 2) = 4

Problem 8: Solve for x: log⁡₈(2x + 1) + log⁡₈(x − 3) = 1

Problem 9: Solve for x: log⁡₉(x² + 3x)=1

Problem 10: Solve for x: log⁡₁₀(x + 7)=3

FAQs on Logarithmic Equations

How do we solve a simple logarithmic equation?

To solve a simple logarithmic equation like log⁡_b(x) = y:

1. Rewrite the equation in its exponential form: x = b^yx = b^yx=by.
2. Solve for x.

How to solve logarithmic equations with multiple logs?

Combine the logarithms using logarithm properties:

1. Use the product, quotient, or power properties to combine logs on each side of the equation.
2. Rewrite the equation in exponential form.
3. Solve for the variable.

Are there any restrictions when solving logarithmic equations?

Yes, the argument of the logarithm must always be positive. So, if you obtain any potential solutions, you must check that they do not make the logarithm of a non-positive number.

What is the common logarithm and the natural logarithm?

• Common Logarithm: Logarithm with base 10, denoted as log⁡₁₀(x) or simply log⁡ (x).
• Natural Logarithm: Logarithm with base e (Euler’s number, approximately 2.718), denoted as ln⁡(x).

How can logarithmic equations be applied in real life?

Logarithmic equations are used in various fields such as:

• Science: To express the pH level in chemistry or the Richter scale for earthquakes.
• Finance: To calculate compound interest and exponential growth/decay.
• Engineering: To measure sound intensity in decibels