https://file.aiquickdraw.com/mathbot/file/17274458775549ar7a1e3.jpg

The image shows a few calculus and logarithmic problems. Let's break them down.

### 1. **Sketch the graphs of the following:**
   **(i)** \( f(x) = 3^{x+4} \)

   This is an exponential function of the form \( 3^x \), but shifted horizontally and vertically due to the \( +4 \) exponent. The function will exhibit:
   - A vertical asymptote as \( x \to -\infty \)
   - Rapid growth as \( x \to +\infty \)
   - It is a transformation of \( 3^x \), shifted 4 units to the left.

   **(ii)** \( h(x) \) (is incomplete in the image, can you clarify?).

### 2. **Sketch the graphs of the following:**
   **(i)** \( f(x) = e^{2x-1} \)

   This is an exponential function with base \( e \). The graph will exhibit:
   - A vertical asymptote as \( x \to -\infty \)
   - Growth at a faster rate than \( e^x \), because of the \( 2x \) term (it stretches horizontally).

   **(ii)** \( h(x) \) (also incomplete in the image, clarification needed).

### 3. **Evaluate the following expressions:**
   **(i)** \( \log \left(\frac{1}{10}\right) \)

   Using logarithmic properties:
   \[
   \log \left(\frac{1}{10}\right) = \log 1 - \log 10 = 0 - 1 = -1
   \]

   **(ii)** \( \log (\sqrt{10}) \)

   Applying properties of logarithms:
   \[
   \log (\sqrt{10}) = \log(10^{1/2}) = \frac{1}{2} \log 10 = \frac{1}{2} \times 1 = 0.5
   \]

Let me know if you need further details or clarifications!

### Expanding Questions:
1. What are the general properties of exponential functions?
2. How do shifts affect the graph of a function like \( f(x) = 3^{x+4} \)?
3. What is the difference between \( e^{2x-1} \) and \( e^{x} \) in terms of graph behavior?
4. What are some common logarithmic properties that help simplify expressions like \( \log \left(\frac{1}{10}\right) \)?
5. How can the logarithmic scale help in solving real-world exponential growth problems?

### Tip:
For graphing transformations, remember that changes in the exponent (horizontal shifts and stretches) dramatically alter the steepness and direction of exponential graphs.

Sketch the graphs of the following functions and evaluate the logarithmic expressions: (i) f(x) = 3^(x+4), f(x) = e^(2x-1), log(1/10), log(√10)

Learn how to graph exponential functions such as f(x) = 3^(x+4) and f(x) = e^(2x-1), and evaluate logarithmic expressions like log(1/10) and log(√10). This exercise covers key concepts in calculus and logarithms.

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