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Chapter 3: Problem 58

Explain why $$ 2-\log x=\log \frac{100}{x} $$ for every positive number \(x\).

Short Answer

Expert verified
The given equation \(2 - \log x = \log \frac{100}{x}\) can be rewritten as \(2 = \log x + \log \frac{100}{x}\). Applying the logarithm property \(\log a + \log b = \log (a*b)\) allows us to simplify the equation to \(2 = \log(100)\). Since \(10^2 = 100\), which is true by definition of logarithm, the original equation holds for every positive number x.

Step by step solution

01

Rewrite the equation

To make the equation more approachable, we can rewrite it by moving all the terms involving logarithms to one side of the equation. $$ 2 = \log x + \log \frac{100}{x} $$
02

Apply logarithm property

We can simplify the logarithmic terms using the logarithm property \(\log a + \log b = \log (a*b)\). We apply this property to the equation from step 1: $$ 2 = \log\left(x * \frac{100}{x}\right) $$
03

Simplify the expression

Next, we will simplify the expression \(x * \frac{100}{x}\): $$ 2 = \log(100) $$
04

Use the definition of logarithm

We know that the logarithm to base 10 is defined as: \(\log_{10} a = b \iff 10^b = a\). Let's use this property to rewrite the equation from step 3: $$ 10^2 = 100 $$ Since this equation is true, it proves that the original equation \(2 - \log x = \log \frac{100}{x}\) is also true for every positive number x.

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