Exponential and Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = a^x is x = a^y. The logarithmic function y = log_ax is defined to be equivalent to the exponential equation x = a^y, y = log_ax, only under the following conditions: x = a^y, a > 0, and a ≠ 1.

Exponential Functions

A function whose value is a constant raised to the power of the argument, especially the function where the constant is e. An exponential function is defined mathematically as follows:

f(x) = a ⋅ b^x

Where,

• a represents a constant that can represent the starting point or the zero point, or the intersection with the y-axis.
• b is the parameter of the exponential function, which is a positive real number, but it can not equal to 1.
• x is denoted as the exponent or the independent variable whenever the case may be.

The base b plays a vital role as it dictates the growth rate of the function. When b > 1, the function demonstrates exponential growth. Conversely, when 0 < b < 1, it reflects exponential decay.

Properties of Exponential Function

• Growth and Decay: In case b > 1, the function f(x) increases rapidly as the value of x increases. On the other hand if, 0 < b < 1, f(x) experiences rapid decreases.
• Intercept: The form of the function is such that the y-intercept is defined by the point (0, a).
• Asymptote: ‘y = 0’ is another asymptote known as the horizontal asymptote because the behavior of the function gets closer to this line but does not cross it as x approaches negative infinity.
• Continuous and Smooth: Exponential functions retain continuity in all the real quantities and do not experience any form of shuffling or skip.

Some examples of exponential function are:

• Compound Interest

The formula of compound interest is A = P (1 + r/n)^nt to find out the total amount of money after t years the amount is given by A, P is the principal amount, r is the annual interest rate or the rate of interest per annum and n is the number of times that interest is compounded per year. In this context, the investment’s growth can be modelled as an exponential function where b = (1 + r/n).

• Radioactive Decay

In population dynamics the equation N(t) = N₀ * e^(-λt) represents how much a given quantity has decayed at time t; N₀ is the quantity at time zero, λ is the decay constant and e is the base of the natural logarithm.

Logarithmic Function

The logarithmic function is the inverse of the exponential function. It has two parts base and value. There are two types of logarithmic function i.e., natural log (ln) and log. It can be mathematically expressed as:

f(x) = log_b(x)

Where:

• b is the base of the logarithm, a positive real number that is not equal to one.
• x is the argument of the logarithm and must be a positive value.

The logarithm log_b(x) addresses the question: "To what exponent must b be raised to yield x?"

Properties of Logarithmic Function

• Domain: The argument x must exceed zero (x > 0).
• Range: The output of the logarithmic function encompasses all real numbers (-∞ < f(x) < ∞).
• Intercept: The logarithmic function intersects at (1, 0), indicating that log_b(1) = 0 for any base b.
• Asymptote: The line defined by x = 0 serves as a vertical asymptote. As x approaches zero from the positive side, the value of the function decreases indefinitely, tending towards negative infinity.
• Continuity and Smoothness: Logarithmic functions maintain continuity for all positive values of x and display a smooth, gradual curve. They are devoid of any breaks or discontinuities within their domain.

Some examples of logarithmic function are:

• Decibel Scale: Sound intensity levels are quantified in decibels through the formula 𝐿 = 10 log₁₀(𝐼/𝐼₀), where 𝐼 denotes the intensity of the sound and 𝐼₀ represents a reference intensity.

• pH Scale: The pH of a solution is determined using the equation pH = −log₁₀[H⁺], where [H⁺] indicates the concentration of hydrogen ions present in the solution.

Relation Between Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverse functions of each other, their interrelationship is defined by the following identities:

logₑ(𝑒ˣ) = x OR \bold{𝑒^(logₑ(x)) = x}

These identities are examples to show that each function can efficiently perform the reversal of the other. For instance:

• To solve 𝑒ˣ = y for x, one can apply the logarithm base 𝑒 to both sides: A second form is x = logₑ(y).
• Conversely, to solve logₑ(x) = y for x, one can exponentiate both sides: That explains why x = 𝑒ʸ.

Solved Examples

Example 1: Solve the equation 2^x+1 = 8.

Solution:

Rewrite the equation in terms of the same base:

2^x+1 = 2³

Since the bases are the same, set the exponents equal:

x + 1 = 3

⇒ x = 3 - 1 =2

Answer: x = 2.

Example 2: Solve the logarithmic equation log₅(x + 2) - log₅(x - 1) = 1.

Solution:

Apply the logarithm property: log_b(A) - log_b(B) = log(A/B):

log₅ (x+2/x-1) = 1

Convert to exponential form:

x+2/x-1 = 5¹ = 5

Multiply both sides by (x - 1):

x + 2 = 5(x - 1)

⇒ x + 2 = 5x - 5

⇒ 7 = 4x

⇒ x = 7/4 = 1.75.

Answer: x = 1.75

Example 3: Solve 5e^2x = 20.

Solution:

Isolate the exponential term:

e^2x = 20/5 = 4

Take the natural logarithm of both sides:

ln(e^2x) = ln(4)

2x = ln(4)

⇒ x = ln(4)/2 ≈ 0.693

Example 4: Solve the equation e^3x = 7.

Solution:

Take the natural logarithm of both sides:

ln(e^3x) = ln(7)

Simplify using the property ln(e^y) = y:

3x = ln(7)

⇒ x = ln(7)/3 ≈ 0.626

Answer: x ≈ 0.626

Example 5: Solve the exponential equation 4^x = 9.

Solution:

Take the natural logarithm of both sides:

ln(4^x) = ln(9)

Use the logarithm property ln(a^b) = b ln(a):

x ln(4) = ln(9)

⇒ x = ln(9)/ln(4) ≈ 1.585

Answer: x ≈ 1.585

Example 6: Solve the equation log₃(x) + log₃(x-2) = 1.

Solution:

Combine the logarithms:

log₃(x(x - 2)) = 1

Convert to exponential form:

x(x - 2) = 3¹ = 3

Factor the quadratic equation:

(x - 3)(x + 1) = 0

The solutions are x = 3 and x = -1. However, x must be positive, so x = 3.

Practice Questions

Q1. Solve the equation: 2^x = 1/16.

Q2. Solve the exponential equation: 5^x-1 = 25.

Q3. Solve the exponential equation: 3^x+2 = 81.

Q4. Solve the logarithmic equation: log₇(2x - 1) = 2.

Q5. Solve for x: e^x/2 = 5.

Q6. Solve for x: ln(x² + 4x) = ln(9).

Q7. Solve the equation: log₄ (x + 1) + log₄ (x - 1) = 2.

Answer Key

1. x = √17​
2. x = 2
3. x = 2ln⁡(5)
4. x = 25
5. x = −4
6. x = − 2 ± √13
7. x = 3