Step 1: Understand the original problem

• Scatterplot shows exponential curve (y = ae^(bx))
• Linear model residuals are curved
• Need to linearize the relationship

Step 2: Why try log(y) vs x? Exponential relationship: y = ae^(bx) Take log of both sides: log(y) = log(a) + bx

This is LINEAR in x!

Step 3: What to look for after transformation If log transformation is appropriate: ✓ Scatterplot of log(y) vs x should be LINEAR ✓ Residual plot should show RANDOM scatter ✓ No curved pattern in residuals

Step 4: How to check

1. Create new variable: y' = log(y)
2. Plot y' vs x (should be linear)
3. Fit regression: ŷ' = b₀ + b₁x
4. Check residual plot (should be random)

Step 5: Interpretation After transformation:

• Can use linear regression on log(y) vs x
• To predict y: ŷ = e^(b₀ + b₁x)
• Or: ŷ = e^(b₀) × e^(b₁x)

Answer: After log transformation, the plot of log(y) vs x should show a LINEAR pattern, and residuals should be randomly scattered with no curve.

Step 1: Identify the relationship Power model: y = ax^b (Example: area = πr², where b = 2)

Step 2: Apply log transformation to BOTH variables Take log of both sides: log(y) = log(a × x^b) log(y) = log(a) + log(x^b) log(y) = log(a) + b·log(x)

Step 3: Recognize linear form Let: Y = log(y), X = log(x), A = log(a) Then: Y = A + bX

This is LINEAR!

Step 4: How to transform

1. Create Y = log(y)
2. Create X = log(x)
3. Plot Y vs X (should be linear)
4. Fit regression: Ŷ = b₀ + b₁X

Step 5: Interpret coefficients After regression:

• b₁ = power (exponent b)
• b₀ = log(a), so a = e^(b₀) or a = 10^(b₀)

To predict original y: ŷ = e^(b₀) × x^(b₁) [if using natural log] ŷ = 10^(b₀) × x^(b₁) [if using log base 10]

Step 1: Identify the original problem Fan-shaped residuals mean:

• Variance increases with x
• Violates constant variance assumption
• Often occurs when y grows exponentially

Step 2: Why log(y) helps with variance When y is exponential or multiplicative:

• Larger y values have larger variability
• Variance proportional to mean
• log transformation STABILIZES variance

Mathematical reason: If y has variance proportional to y²: Var(y) ∝ y²

Then: Var(log(y)) ≈ constant (Delta method from calculus)

Step 3: Additional benefit Log transformation often: ✓ Linearizes exponential relationships ✓ Stabilizes variance (fixes fan shape) ✓ Makes distribution more symmetric ✓ Reduces impact of outliers

Step 4: When to use log transformation Use log(y) when you see:

Step 1: Understand the model Fitted equation: log(y) = 2 + 0.5x This uses NATURAL LOG (ln)

Step 2: Predict log(y) for x = 10 log(y) = 2 + 0.5(10) log(y) = 2 + 5 log(y) = 7

Step 3: Back-transform to get y Since we used natural log (ln): ln(y) = 7

To solve for y, use exponential: y = e^7

Step 4: Calculate y = e^7 ≈ 1,096.63

Step 5: Interpretation "When x = 10, y is predicted to be approximately 1,097."

Important notes:

• Must back-transform using e^(predicted value)
• If using log₁₀, would use 10^(predicted value)
• Always specify which log was used!

Alternative form: Original model: y = e^(2 + 0.5x) = e² × e^(0.5x) y = e² × e^(0.5x) ≈ 7.39 × e^(0.5x)

When x = 10: y = 7.39 × e^5 ≈ 1,097

Answer: y = e^7 ≈ 1,097

Step 1: Identify TWO problems

1. Curvature → nonlinear relationship
2. Fan shape → non-constant variance

Need transformation that fixes BOTH!

Step 2: Try log(y) vs x Often works for:

• Exponential relationships (fixes curve)
• Multiplicative error (fixes fan)
• Right-skewed data

Check result: ✓ Should be linear ✓ Should have constant variance

Step 3: If log(y) doesn't work completely Try other transformations:

• √y vs x (square root)
• 1/y vs x (reciprocal)
• log(y) vs log(x) (both sides)

Step 4: Systematic approach