- Combine the first two logarithmic terms using the property \(\log_b(x) - \log_b(y) = \log_b(x/y)\).
- Simplify the resulting fraction \(\frac{10}{35}\) to \(\frac{2}{7}\).
- Use the addition property \(\log_b(x) + \log_b(y) = \log_b(xy)\) to combine with the third term: \(\log_2\left(\frac{2}{7} \times 112\right)\).
- Evaluate the final logarithm \(\log_2(32) = 5\) to reach the final answer of \(\boxed{5}\).
Explanation
- Analysis and Problem Overview
We are given the expression \(\log_2(10) - \log_2(35) + \log_2(112)\). Our objective is to evaluate this expression using the fundamental Laws of Logarithms. Recall the following properties:
- \(\log_b(x) + \log_b(y) = \log_b(xy)\)
- \(\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\)
- \(\log_b(b^n) = n\)
We will use these laws systematically to combine the terms and simplify the expression into a single number.
- Combining the First Two Terms
First, we combine the first two terms using the subtraction rule for logarithms, \(\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\):
\[\log_2(10) - \log_2(35) = \log_2\left(\frac{10}{35}\right)\]
We can simplify the fraction \(\frac{10}{35}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
\[\frac{10}{35} = \frac{10 \div 5}{35 \div 5} = \frac{2}{7}\]
So, the expression becomes \(\log_2\left(\frac{2}{7}\right) + \log_2(112)\).
- Combining and Simplifying the Full Expression
Now, we apply the addition rule for logarithms, \(\log_b(x) + \log_b(y) = \log_b(xy)\), to combine the remaining terms:
\[\log_2\left(\frac{2}{7}\right) + \log_2(112) = \log_2\left(\frac{2}{7} \times 112\right)\]
Next, we calculate the product inside the logarithm:
\[\frac{2}{7} \times 112 = 2 \times \left(\frac{112}{7}\right) = 2 \times 16 = 32\]
Substituting this value back into the expression, we get:
\[\log_2(32)\]
- Final Evaluation
Finally, we evaluate \(\log_2(32)\). We know that \(32\) can be written as a power of 2:
\[32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\]
Applying the property \(\log_b(b^n) = n\), we have:
\[\log_2(2^5) = 5\]
Thus, the value of the given expression is 5. We have successfully evaluated the expression using the laws of logarithms. \(\boxed{5}\)
Examples
Logarithms are used extensively in fields like acoustics to measure sound intensity in decibels (dB), where sound intensity is compared on a logarithmic scale. Understanding how to manipulate logarithmic expressions allows engineers to simplify complex signal processing calculations and represent vast ranges of data in manageable, readable values. This makes logarithmic identities a fundamental tool for data analysis in science, finance, and engineering.