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Chapter 3: Problem 58

Explain why $$ \log 500=3-\log 2 $$

Short Answer

Expert verified
In short, the given equation \(\log 500 = 3 - \log 2\) is correct because we can rearrange it and apply logarithm properties to obtain an equality. Specifically, we rewrite 3 as \(\log 1000\), which leads to \(\log 500 = \log 1000 - \log 2\). By applying the product rule for logarithms, we combine the right-hand side and simplify to get \(\log 500 = \log 500\), confirming the equation to be true.

Step by step solution

01

Remember the Product Rule

The product rule says that if two numbers have the same base, the logarithm of their product is the sum of their individual logarithms. In general, for any base \(b\), if \(x > 0\) and \(y > 0\), then: \[ \log_b(xy) = \log_b(x) + \log_b(y) \]
02

Write the Right-hand side in a Different Form

We are given: \[ \log 500 = 3 - \log 2 \] Now, rewrite 3 as a logarithm with base 10 and apply the product rule: \[ 3 = \log 10^3 = \log 1000 \] So, \[ \log 500 = \log 1000 - \log 2 \]
03

Apply the Product Rule

Now we use the product rule backward to combine the right-hand side: \[ \log 500 = \log(1000 / 2) \]
04

Simplify

1000 / 2 is equal to 500. So the equation becomes: \[ \log 500 = \log 500\] This equation is true, so the given equation is correct. The reason is that we could rearrange the equation and apply the logarithm's properties to get an equality.

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