14) Use logarithms to express the following equations in the form $$ y=m x+c $$ and state the gradients of each line (a) $$ P=6 x^{5} $$ (b) $$ y=5 e^{3 x} $$

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To express the given equations in the linear form $$ y = m x + c $$ using logarithms and determine the gradients ($$ m $$), follow the steps below for each part:

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### **(a) $$ P = 6 x^{5} $$**

**Step 1: Take the natural logarithm of both sides.**
$$
\ln P = \ln(6 x^{5})
$$

**Step 2: Use logarithm properties to simplify.**
$$
\ln P = \ln 6 + \ln x^{5} = \ln 6 + 5 \ln x
$$

**Step 3: Rearrange to match the linear form $$ Y = m X + c $$.**
$$
\ln P = 5 \ln x + \ln 6
$$
Here, if we let:
- $$ Y = \ln P $$
- $$ X = \ln x $$
  
Then the equation becomes:
$$
Y = 5X + \ln 6
$$

**Gradient:**
$$
m = 5
$$

---

### **(b) $$ y = 5 e^{3 x} $$**

**Step 1: Take the natural logarithm of both sides.**
$$
\ln y = \ln(5 e^{3x})
$$

**Step 2: Use logarithm properties to simplify.**
$$
\ln y = \ln 5 + \ln e^{3x} = \ln 5 + 3x
$$

**Step 3: Rearrange to match the linear form $$ Y = m X + c $$.**
$$
\ln y = 3x + \ln 5
$$
Here, if we let:
- $$ Y = \ln y $$
- $$ X = x $$
  
Then the equation becomes:
$$
Y = 3X + \ln 5
$$

**Gradient:**
$$
m = 3
$$

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### **Summary of Results**

- **(a)** After taking logarithms, the equation becomes:
  $$
  \ln P = 5 \ln x + \ln 6
  $$
  **Gradient:** $$ m = 5 $$

- **(b)** After taking logarithms, the equation becomes:
  $$
  \ln y = 3x + \ln 5
  $$
  **Gradient:** $$ m = 3 $$

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These linear forms allow you to plot $$ \ln P $$ versus $$ \ln x $$ for part (a) and $$ \ln y $$ versus $$ x $$ for part (b), with the gradients representing the coefficients of $$ \ln x $$ and $$ x $$, respectively.
To express the equations in the form $$ y = m x + c $$ using logarithms and find the gradients: **(a) $$ P = 6 x^{5} $$** - Take logs: $$ \ln P = 5 \ln x + \ln 6 $$ - Gradient ($$ m $$): 5 **(b) $$ y = 5 e^{3 x} $$** - Take logs: $$ \ln y = 3x + \ln 5 $$ - Gradient ($$ m $$): 3

14) Use logarithms to express the following equations in the form \( y=m x+c \) and state the gradients of each line (a) \( P=6 x^{5} \) (b) \( y=5 e^{3 x} \)

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