To express the exponential function $y = 33 \times 10^{-0.9x}$ in the form of a linear function in a logarithmic scale, we can proceed as follows:

Step 1: Take the logarithm of both sides

We typically use the logarithm with base 10 (common logarithm) for logarithmic scales.

$\log_{10}(y) = \log_{10}\left(33 \times 10^{-0.9x}\right)$

Step 2: Apply the logarithm properties

Use the properties of logarithms to simplify:

$\log_{10}(y) = \log_{10}(33) + \log_{10}\left(10^{-0.9x}\right)$

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

Since $\log_{10}(10) = 1$, we simplify further:

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

Step 3: Express the equation in the linear form

Let $Y = \log_{10}(y)$ and $X = x$. The equation becomes:

$Y = -0.9X + \log_{10}(33)$

This is the linear form of the function in the logarithmic scale, where:

• The slope $m$ is $-0.9$.
• The y-intercept $b$ is $\log_{10}(33)$.

Thus, the linear function is:

$\log_{10}(y) = -0.9x + \log_{10}(33)$

Would you like more details or have any questions?

Related Questions:

1. How do you convert other exponential functions to linear form in a logarithmic scale?
2. What are the logarithm properties used in simplifying expressions?
3. How do you determine the slope and intercept from an exponential function?
4. What are common applications of expressing functions in logarithmic scales?
5. How does the base of the logarithm affect the transformation?

Tip: When converting exponential functions to linear form, always remember to apply the logarithm to the entire function, and use logarithm properties to simplify.