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Chapter 3: Problem 71

Condense the expression to the logarithm of a single quantity. $$2 \log _{2} x+4 \log _{2} y$$

Short Answer

Expert verified
The condensed form of the given expression is \(\log _{2} (x^2y^4)\)

Step by step solution

01

Apply the Power Rule

The Power Rule states that the logarithm of a number raised to a power is equal to the product of the log base and the exponent. This rule is applied to get: \[2 \log _{2} x+4 \log _{2} y = \log _{2} x^2 + \log _{2} y^4\]
02

Use the Product Rule

The Product Rule states that the sum of the logarithms of two numbers is the same as the logarithm of their product. Apply this rule to get: \[\log _{2} x^2 + \log _{2} y^4 = \log _{2} (x^2y^4)\]

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Most popular questions from this chapter

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

Rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Find the \(\mathrm{pH}\) if \(\left[\mathrm{H}^{+}\right]=2.3 \times 10^{-5}\).

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Apple juice has a pH of 2.9 and drinking water has a pH of \(8.0 .\) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?

The populations \(P\) (in thousands) of Reno, Nevada from 2000 through 2007 can be modeled by \(P=346.8 e^{k t},\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . In \(2005,\) the population of Reno was about 395,000 . (Source: U.S. Census Bureau) (a) Find the value of \(k\). Is the population increasing or decreasing? Explain. (b) Use the model to find the populations of Reno in 2010 and 2015 . Are the results reasonable? Explain. (c) According to the model, during what year will the population reach \(500,000 ?\)
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