Write down the exponential function of the form y=33×10−0.9x in the form of a linear function in a logarithmic scale.

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.

Learn how to convert the exponential function y = 33 * 10^(-0.9x) into a linear form using logarithmic scales. This problem covers exponential functions, logarithmic properties, and their application in transforming functions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation