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Chapter 33: Problem 958

Find \(\log _{10}\left(10^{2} \cdot 10^{-3} \cdot 10^{5}\right)\)

Short Answer

Expert verified
The short answer is: \(\log_{10}\left(10^{2} \cdot 10^{-3} \cdot 10^{5}\right) = 4\).

Step by step solution

01

1. Rewrite the expression using properties of logarithms

Recall the properties of logarithms: $$\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)$$ $$\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$$ Using the first property of logarithms, we can rewrite our expression: $$\log_{10}\left(10^{2} \cdot 10^{-3} \cdot 10^{5}\right) = \log_{10}(10^{2}) + \log_{10}(10^{-3}) + \log_{10}(10^{5})$$
02

2. Evaluate each logarithm

The property we'll use here is: $$\log_{b}(b^{x}) = x$$ Applying this property to each log in the sum: $$\log_{10}(10^{2}) + \log_{10}(10^{-3}) + \log_{10}(10^{5}) = 2 + (-3) + 5$$
03

3. Simplify the expression

Now, sum the values inside the expression: $$2 + (-3) + 5 = - 1 + 5 = 4$$ So, the final answer is: $$\log_{10}\left(10^{2} \cdot 10^{-3} \cdot 10^{5}\right) = 4$$

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