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Chapter 3: Problem 58

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. $$2 \ln x-\frac{1}{2} \ln y$$

Short Answer

Expert verified
The simplified expression of the given logarithmic function is \(\ln\left(\frac{x^2}{y^{1/2}}\right)\)

Step by step solution

01

Transform coefficients into exponents

We transform each coefficient into the exponent of the argument of the corresponding log or ln. Giving us \(\ln x^2 - \ln y^{1/2}\)
02

Combine Logarithms

We use the property that the difference of two logs equals the log of the quotient, so now we rewrite the expression as a single logarithm \(\ln\left(\frac{x^2}{y^{1/2}}\right)\)
03

Simplify

If it's possible, we simplify the expression under the log. In this case, nothing more can be simplified.

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Most popular questions from this chapter

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ a. Use your graphing utility's exponential regression option to obtain a model of the form \(y=a b^{x}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data? b. Rewrite the model in terms of base \(e\). By what percentage is the population of the United States increasing each year?

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