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The image contains a list of logarithmic laws and three equations where you're asked to express each in the form \( y = mx + c \), and also determine the gradients (slopes) of the lines.

### Laws of Logarithms (as seen in the image):
1. \( \log_a b^n = n \log_a b \)
2. \( \log_a b + \log_a c = \log_a (bc) \)
3. \( \log_a b - \log_a c = \log_a \left(\frac{b}{c}\right) \)

### Given Equations:
1. \( y = ab^x \)
2. \( y = ax^n \)
3. \( f = \frac{c}{2\pi} \sqrt{\frac{A}{LV}} \)

#### To solve:

1. **First Equation: \( y = ab^x \)**
   - Take the natural logarithm (ln) of both sides:
     \[
     \ln y = \ln(ab^x) = \ln a + \ln(b^x) = \ln a + x \ln b
     \]
   - This matches the form \( y = mx + c \), where:
     \[
     m = \ln b, \quad c = \ln a
     \]
   - The gradient (slope) is \( \ln b \).

2. **Second Equation: \( y = ax^n \)**
   - Take the natural logarithm of both sides:
     \[
     \ln y = \ln(ax^n) = \ln a + \ln(x^n) = \ln a + n \ln x
     \]
   - This is in the form \( y = mx + c \), where:
     \[
     m = n, \quad c = \ln a
     \]
   - The gradient (slope) is \( n \).

3. **Third Equation: \( f = \frac{c}{2\pi} \sqrt{\frac{A}{LV}} \)**
   - To express this equation in the form \( y = mx + c \), we would first simplify it:
     \[
     f = \frac{c}{2\pi} \sqrt{\frac{A}{LV}}
     \]
   - This one doesn't involve a logarithmic transformation directly but can be simplified further depending on what the variables represent. More details might be needed to determine how to express it in slope-intercept form.

Would you like me to dive into the third equation further or explore related questions?

### Follow-up Questions:
1. How do we determine the logarithmic forms of more complex expressions?
2. Why is taking the logarithm useful for linearization of exponential functions?
3. What are the common logarithmic rules used in simplifying complex equations?
4. How do logarithmic transformations help in finding gradients in exponential equations?
5. How can we apply these transformations in solving real-world growth or decay problems?

**Tip:** When handling equations of the form \( y = ab^x \), taking the logarithm helps convert the exponential relationship into a linear form, simplifying analysis and interpretation.

Use the laws of logarithms to express each of the following equations in the form y = mx + c. State also the gradients of each of the lines. The given equations are: y = ab^x, y = ax^n, and f = (c / 2π) √(A / LV).

This exercise explores how to apply the laws of logarithms to express exponential equations in the linear form y = mx + c. The solutions involve transforming equations like y = ab^x and y = ax^n, using logarithmic rules and stating the gradients (slopes) of the resulting linearized equations.

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