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Chapter 3: Problem 63

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\frac{1}{2}(\log x+\log y)$$

Short Answer

Expert verified
The condensed single logarithm of the expression is \(\log(\sqrt{xy})\).

Step by step solution

01

Distribute the Fraction

Use the rule \(\log(mn) = \log(m) + \log(n)\) in reverse. Begin by distributing the \(\frac{1}{2}\) to both \(\log(x)\) and \(\log(y)\) so you get \(\frac{1}{2} \log(x) + \frac{1}{2} \log(y)\)
02

Apply Exponent rule of Logarithms

For each term, apply \(n \log(m) = \log(m^n)\) to put the \(\frac{1}{2}\) as the exponent of the argument of the logarithms, like so: \(\log(x^{1/2}) + \log(y^{1/2})\) which evaluates to \(\log(\sqrt{x}) + \log(\sqrt{y})\)
03

Condense the Logarithms

Use the rule \(\log(mn) = \log(m) + \log(n)\) to condense the two logs into one by multiplying 'm' and 'n' (which are \(\sqrt{x}\) and \(\sqrt{y}\) in this case). The result will be \(\log(\sqrt{x}*\sqrt{y})\)
04

Simplify the Expression

Simplify the expression inside the logarithm. The multiplication of two square roots is equal to the square root of the multiplication of the two initial numbers. So, \(\sqrt{x}*\sqrt{y}\) evaluates to \(\sqrt{xy}\). Thus, you have \(\log(\sqrt{xy})\)

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Most popular questions from this chapter

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