https://file.aiquickdraw.com/mathbot/file/17254676553139wvg35g2.jpg

The expression in the image is:

\[
\log \left(\frac{10}{10^m}\right)
\]

Let's simplify this expression:

1. First, recognize that the expression inside the logarithm is a fraction where the numerator is \(10\) and the denominator is \(10^m\). This can be rewritten using the properties of exponents:

\[
\frac{10}{10^m} = 10^{1-m}
\]

2. Now, the expression becomes:

\[
\log \left(10^{1-m}\right)
\]

3. Using the logarithm power rule, \(\log(a^b) = b \cdot \log(a)\), where \(a = 10\) and \(b = 1-m\):

\[
\log \left(10^{1-m}\right) = (1-m) \cdot \log(10)
\]

4. Since \(\log(10)\) in base 10 is equal to 1, the expression simplifies further:

\[
1 - m
\]

So, the simplified result of the expression \(\log \left(\frac{10}{10^m}\right)\) is:

\[
\boxed{1 - m}
\]

Would you like to explore further details or have any questions?

Here are 5 related questions to consider:
1. How would this logarithmic expression change if the base of the logarithm were different?
2. What would be the result if \(m\) were a negative number?
3. Can you apply similar logarithmic rules to expressions involving other bases or constants?
4. How would you handle a logarithmic equation where the argument is a product instead of a quotient?
5. What is the interpretation of this logarithmic expression in a real-world context?

**Tip:** Always remember that the logarithm of a number in its own base is 1, which simplifies many logarithmic expressions quickly!

Explore how to simplify the logarithmic expression log(10 / 10^m) using the logarithm power rule. Learn the step-by-step process to arrive at the simplified result, 1 - m, suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation