Write down the exponential function of the form  $$ y=\text{ }33 $$×$$ 10^{-0.9x} $$

in the form of a linear function in a logarithmic scale.

Round off the numbers in the formula to one decimal place.

To convert the exponential function $ y = 33 \times 10^{-0.9x} $ into a linear function in a logarithmic scale, we'll follow these steps:

1. **Take the logarithm (base 10) of both sides:**
   \[
   \log_{10}(y) = \log_{10}\left(33 \times 10^{-0.9x}\right)
   \]

2. **Use logarithm properties:**
   \[
   \log_{10}(y) = \log_{10}(33) + \log_{10}\left(10^{-0.9x}\right)
   \]

   \[
   \log_{10}(y) = \log_{10}(33) - 0.9x
   \]

3. **Calculate $\log_{10}(33)$:**
   \[
   \log_{10}(33) \approx 1.5
   \]

4. **Substitute back into the equation:**
   \[
   \log_{10}(y) \approx 1.5 - 0.9x
   \]

So, the linear form of the given exponential function on a logarithmic scale is:

\[
\log_{10}(y) = 1.5 - 0.9x
\]

This represents a straight line in a logarithmic plot, where the slope is $-0.9$ and the intercept is $1.5$.

Would you like further details or have any questions?

Here are 5 related questions:
1. How does changing the base of the logarithm affect the linear representation?
2. What is the significance of the slope $-0.9$ in the logarithmic scale?
3. How can you determine the original function from the linear logarithmic form?
4. What would the graph of $ \log_{10}(y) = 1.5 - 0.9x $ look like?
5. How would you approach this problem using natural logarithms instead of common logarithms?

**Tip:** Remember, taking the logarithm of an exponential function linearizes the expression, making it easier to analyze relationships like growth and decay.

Learn how to convert the exponential function y = 33 * 10^(-0.9x) into a linear form in a logarithmic scale. Understand the logarithm base 10 transformation, logarithm properties, and the resulting linear equation. Suitable for advanced high school students and beyond.

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