Topic 6 of 17Cambridge A Levels

Logarithmic and Exponential Functions

Explore inverse functions e^x and ln x, their graphs, properties, and use in solving equations.

### Introduction to Exponential and Logarithmic Functions

In mathematics, we often study pairs of functions that are inverses of each other; one function 'undoes' what the other does. The natural exponential function, denoted as f(x) = e^x, and the natural logarithmic function, denoted as g(x) = ln x, form such a pair. These functions are fundamental in modelling various real-world phenomena, from population growth to radioactive decay.

The constant 'e' is a special irrational number, approximately equal to 2.71828. It is also known as Euler's number and is the base of the natural logarithm.

### The Exponential Function: y = e^x

The function y = e^x describes exponential growth. Understanding its graph and properties is crucial.

Key Properties of y = e^x:

Domain: The function is defined for all real numbers x (-∞, ∞).

Range: The output, e^x, is always positive. The range is (0, ∞).

Y-intercept: When x = 0, y = e^0 = 1. The graph always passes through the point (0, 1).

Asymptote: As x approaches -∞, e^x approaches 0. The x-axis (the line y = 0) is a horizontal asymptote.

Monotonicity: The function is strictly increasing; as x increases, y increases at an accelerating rate.

### The Natural Logarithmic Function: y = ln x

The natural logarithm is the inverse of the exponential function. The statement y = ln x is equivalent to x = e^y. In simple terms, ln x is the power to which 'e' must be raised to get x.

Key Properties of y = ln x:

Domain: The logarithm is only defined for positive numbers. The domain is (0, ∞).

Range: The output can be any real number. The range is (-∞, ∞).

X-intercept: When y = 0, x = e^0 = 1. The graph always passes through the point (1, 0).

Asymptote: As x approaches 0 from the positive side, ln x approaches -∞. The y-axis (the line x = 0) is a vertical asymptote.

Inverse Relationship: The graph of y = ln x is a reflection of the graph of y = e^x in the line y = x.

### Laws of Logarithms

To manipulate and solve equations involving logarithms, we use three fundamental laws. For any positive numbers a and b, and any real number n:

Product Rule: ln(ab) = ln a + ln b

Quotient Rule: ln(a/b) = ln a - ln b

Power Rule: ln(a^n) = n ln a

Additionally, two key identities arise from the inverse relationship:

* ln(e) = 1 (since e^1 = e)

* ln(1) = 0 (since e^0 = 1)

* e^(ln x) = x and ln(e^x) = x

### Solving Equations

The primary application of these functions in A Level Mathematics is solving equations.

1. Solving Exponential Equations

The core strategy is to take the natural logarithm of both sides to bring the variable down from the exponent.

*Example:* Solve e^(2x+1) = 10.

Process:

Take the natural log of both sides: ln(e^(2x+1)) = ln(10)

Use the property ln(e^a) = a: 2x + 1 = ln(10)

Isolate x: 2x = ln(10) - 1

x = (ln(10) - 1) / 2

Sometimes, an equation can be reduced to a quadratic form.

*Example:* Solve e^(2x) - 6e^x + 8 = 0.

Process:

Notice that e^(2x) = (e^x)^2. Let y = e^x.

The equation becomes a quadratic: y^2 - 6y + 8 = 0

Factorise the quadratic: (y - 4)(y - 2) = 0

So, y = 4 or y = 2.

Substitute back e^x for y:

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

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

2. Solving Logarithmic Equations

The main strategy is to combine logarithms using the laws and then rewrite the equation in exponential form.

*Example:* Solve ln(x - 3) = 2.

Process:

Rewrite in exponential form (if ln a = b, then a = e^b): x - 3 = e^2

Isolate x: x = e^2 + 3

*Example:* Solve 2 ln(x) - ln(x - 1) = ln(4).

Process:

Use the power rule: ln(x^2) - ln(x - 1) = ln(4)

Use the quotient rule: ln(x^2 / (x - 1)) = ln(4)

If ln A = ln B, then A = B: x^2 / (x - 1) = 4

Solve the resulting equation: x^2 = 4(x - 1) => x^2 = 4x - 4 => x^2 - 4x + 4 = 0

Factorise: (x - 2)^2 = 0 => x = 2

Crucial Step: Always **check the solution**. The original equation has ln(x) and ln(x-1). For x=2, both arguments (2 and 2-1=1) are positive, so the solution is valid.

Key Points to Remember

1The functions y = e^x and y = ln x are inverses of each other.
2The graph of y = ln x is a reflection of y = e^x in the line y = x.
3Key graph points: y = e^x passes through (0, 1); y = ln x passes through (1, 0).
4Master the three laws of logarithms: Product Rule, Quotient Rule, and Power Rule.
5To solve exponential equations like e^(ax) = b, take the natural log of both sides.
6To solve logarithmic equations like ln(ax) = b, rewrite them in exponential form (ax = e^b).
7Equations like e^(2x) + A*e^x + B = 0 can be solved by substituting y = e^x to form a quadratic.
8Always verify solutions for logarithmic equations to ensure the arguments remain positive.

Pakistan Example

Modelling Pakistan's Population Growth

Exponential functions are used to model population growth. The formula is P(t) = P₀ * e^(rt), where P(t) is the population after t years, P₀ is the initial population, and r is the annual growth rate. **Problem:** In 2020, Pakistan's population was approximately 220 million. Assuming a constant annual growth rate of 1.9% (r = 0.019), in which year will the population reach 300 million? **Solution:** 1. Set up the equation: 300 = 220 * e^(0.019t) 2. Isolate the exponential term: 300 / 220 = e^(0.019t) => 1.3636 ≈ e^(0.019t) 3. Take the natural log of both sides: ln(1.3636) = ln(e^(0.019t)) 4. Simplify: 0.3102 ≈ 0.019t 5. Solve for t: t ≈ 0.3102 / 0.019 ≈ 16.3 years. **Conclusion:** The population is projected to reach 300 million approximately 16.3 years after 2020, which would be during the year 2036. This demonstrates how logarithms are essential for solving for time in exponential growth models.

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Mathematics — Quick Revision

Logarithmic and Exponential Functions

Key Concepts

1The functions y = e^x and y = ln x are inverses of each other.

2The graph of y = ln x is a reflection of y = e^x in the line y = x.

3Key graph points: y = e^x passes through (0, 1); y = ln x passes through (1, 0).

4Master the three laws of logarithms: Product Rule, Quotient Rule, and Power Rule.

5To solve exponential equations like e^(ax) = b, take the natural log of both sides.

6To solve logarithmic equations like ln(ax) = b, rewrite them in exponential form (ax = e^b).

Formulas to Know

The functions y = e^x and y = ln x are inverses of each other.

The graph of y = ln x is a reflection of y = e^x in the line y = x.

Key graph points: y = e^x passes through (0, 1); y = ln x passes through (1, 0).

To solve exponential equations like e^(ax) = b, take the natural log of both sides.

Pakistan Example

Modelling Pakistan's Population Growth

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