Grade: Grade 11 Subject: Mathematics Unit: Functions & Modeling SAT: AdvancedMath ACT: Math

Exponential and Logarithmic

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Exponential Functions Review

Exponential Function

f(x) = a · bˣ

  • a = initial value (y-intercept when x = 0)
  • b = base (growth/decay factor)
  • b > 1: exponential growth
  • 0 < b < 1: exponential decay

The Natural Base e

The number e ≈ 2.71828... is called the natural base. It appears naturally in continuous growth situations.

Continuous growth/decay: A(t) = A₀ · eᵏᵗ

  • k > 0: continuous growth
  • k < 0: continuous decay

Logarithmic Functions

Definition of Logarithm

A logarithm is the inverse of an exponential function:

y = logᵦ(x) means bʸ = x

"log base b of x equals y" means "b raised to the y power equals x"

Common Logarithms

  • Common log: log(x) = log₁₀(x) (base 10)
  • Natural log: ln(x) = logₑ(x) (base e)

Properties of Logarithms

Logarithm Rules

RuleFormula
Product Rulelogᵦ(MN) = logᵦ(M) + logᵦ(N)
Quotient Rulelogᵦ(M/N) = logᵦ(M) - logᵦ(N)
Power Rulelogᵦ(Mⁿ) = n · logᵦ(M)
Change of Baselogᵦ(x) = log(x)/log(b) = ln(x)/ln(b)
Identitylogᵦ(b) = 1
Identitylogᵦ(1) = 0
Inverseb^(logᵦ(x)) = x
Inverselogᵦ(bˣ) = x

Solving Exponential Equations

Method 1: Same Base

If bˣ = bʸ, then x = y

Method 2: Taking Logarithms

If bˣ = c, then x = logᵦ(c) = ln(c)/ln(b)

Solving Logarithmic Equations

  1. Isolate the logarithm on one side
  2. Convert to exponential form: if logᵦ(x) = y, then x = bʸ
  3. Check for extraneous solutions (log argument must be positive)

Growth and Decay Models

Exponential Growth/Decay

A(t) = A₀(1 + r)ᵗ (discrete)

A(t) = A₀ · eʳᵗ (continuous)

Where:

  • A₀ = initial amount
  • r = rate (positive for growth, negative for decay)
  • t = time

Half-Life and Doubling Time

Half-life: Time for quantity to reduce by half

t½ = ln(2)/k ≈ 0.693/k

Doubling time: Time for quantity to double

t₂ = ln(2)/k ≈ 0.693/k

Examples

Example 1: Expanding Logarithms

Problem: Expand: log₃(x²y/z)

Solution:

Step 1: Apply quotient rule

= log₃(x²y) - log₃(z)

Step 2: Apply product rule

= log₃(x²) + log₃(y) - log₃(z)

Step 3: Apply power rule

= 2log₃(x) + log₃(y) - log₃(z)

Example 2: Condensing Logarithms

Problem: Write as a single logarithm: 3ln(x) - ln(y) + ½ln(z)

Solution:

Step 1: Apply power rule (in reverse)

= ln(x³) - ln(y) + ln(z^½)

= ln(x³) - ln(y) + ln(√z)

Step 2: Apply product and quotient rules

= ln(x³√z/y)

Example 3: Solving Exponential Equations

Problem: Solve: 5ˣ⁺² = 125

Solution:

Step 1: Write 125 with base 5

5ˣ⁺² = 5³

Step 2: Since bases are equal, exponents are equal

x + 2 = 3

x = 1

Example 4: Solving with Logarithms

Problem: Solve: 3ˣ = 20

Solution:

Step 1: Take natural log of both sides

ln(3ˣ) = ln(20)

Step 2: Apply power rule

x · ln(3) = ln(20)

Step 3: Solve for x

x = ln(20)/ln(3) ≈ 2.996/1.099 ≈ 2.727

Example 5: Population Growth

Problem: A population of bacteria doubles every 4 hours. If there are initially 500 bacteria, how many will there be after 10 hours?

Solution:

Step 1: Find the growth rate using doubling time

4 = ln(2)/k → k = ln(2)/4 ≈ 0.173

Step 2: Use continuous growth formula

A(t) = 500 · e^(0.173 × 10)

A(10) = 500 · e^1.73

A(10) ≈ 500 × 5.64 ≈ 2,828 bacteria

Practice

1. Expand: log₂(8x³/y)

2. Condense: 2log(x) + log(y) - 3log(z)

3. Solve: 4ˣ = 64

4. Solve: 2ˣ⁻¹ = 7

5. Solve: log₃(x + 2) = 4

6. Solve: ln(x) + ln(x - 2) = ln(15)

7. Evaluate without a calculator: log₄(64)

8. Use change of base to find: log₅(17)

9. A radioactive substance has a half-life of 10 years. If you start with 80 grams, how much remains after 25 years?

10. An investment of $1000 grows at 5% annual interest compounded continuously. When will it reach $1500?

Click to reveal answers
  1. 3 + 3log₂(x) - log₂(y)
  2. log(x²y/z³)
  3. x = 3
  4. x = 1 + log₂(7) ≈ 3.807
  5. x = 79
  6. x = 5 (check: ln(5) + ln(3) = ln(15) ✓)
  7. 3 (since 4³ = 64)
  8. ln(17)/ln(5) ≈ 1.76
  9. 80(0.5)^(25/10) = 80(0.5)^2.5 ≈ 14.14 grams
  10. 1500 = 1000e^(0.05t) → t = ln(1.5)/0.05 ≈ 8.11 years

Check Your Understanding

1. Why is the argument of a logarithm always positive?

Show answer

Since logᵦ(x) asks "what power of b gives x?", and any positive base b raised to any real power always gives a positive result, x must be positive. There's no real number y such that bʸ = 0 or bʸ = negative.

2. Explain why ln(eˣ) = x and e^(ln x) = x.

Show answer

These are inverse function properties. ln(eˣ) = x because ln "undoes" the exponential. e^(ln x) = x because the exponential "undoes" the logarithm. They are reflections of each other over y = x.

3. What's the difference between exponential growth at rate r and continuous exponential growth at rate r?

Show answer

Discrete: A = A₀(1+r)ᵗ compounds at fixed intervals. Continuous: A = A₀eʳᵗ compounds infinitely often. Continuous growth is always slightly faster because interest/growth is constantly being added to the base amount.

4. Why does the half-life formula only depend on k, not on the initial amount?

Show answer

Half-life is when A(t) = A₀/2. Solving A₀e^(-kt) = A₀/2, the A₀ cancels: e^(-kt) = 1/2. So t = ln(2)/k. The proportion that decays in a given time is independent of how much you started with.

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