📝 Chapter 7: Solving Exp. & Log. Equations

Solve exactly: a) 2³ˣ⁻¹ = 16 b) 9ˣ = 27 c) (1/4)ˣ = 8

Solve to 3 decimal places: a) 5ˣ = 12 b) 3²ˣ⁺¹ = 50

Solve: a) log₄(x) = 3/2 b) log(2x+5) = 1 c) ln(x²) = 4

Solve and check: log₂(x+3) + log₂(x-1) = 3

Solve: 2ˣ·4ˣ⁺¹ = 8³ (hint: express all in base 2)

Solve: 3ˣ⁺¹ - 3ˣ = 162. Hint: factor out 3ˣ.

Find all solutions: (log x)² - log(x²) = 3. Careful — this is a quadratic in disguise.

How many solutions does 2ˣ = x² have? Justify your answer graphically and explain why algebraic methods alone are insufficient.

A student solved log₃(x-2) = log₃(5) + log₃(x) and got x = -2/4. Identify the error and solve correctly.

Write a complete step-by-step solution guide for solving exponential equations of the form aˣ = b, including when you can match bases vs when you must use logarithms.

Explain the concept of extraneous solutions in logarithmic equations. Why do they occur? How do you identify them? Use a specific example.

Compare solving 2ˣ = 8 vs 2ˣ = 7. How are the strategies different? Which requires logarithms and why?

Create a decision flowchart for a student facing an equation with exponents or logs. The flowchart should help them decide which strategy to use.

A bacteria population doubles every 3 hours. Starting with 100 bacteria: a) Write an exponential model. b) When will there be 1 million bacteria? c) What is the hourly growth rate?

Carbon-14 has a half-life of 5730 years. A fossil has 15% of its original C-14. How old is the fossil?

An investment of $5000 earns 4.5% compounded monthly: A = 5000(1 + 0.045/12)¹²ᵗ. When will it reach $8000?

Two towns have populations P₁(t) = 25000·(1.03)ᵗ and P₂(t) = 40000·(1.01)ᵗ. When will their populations be equal?