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Chapter 4: Problem 59

$$\text { Find each value. If applicable, give an approximation to four decimal places.}$$ $$\ln 98-\ln 13$$

Short Answer

Expert verified
2.0212

Step by step solution

01

Apply the properties of logarithms

Use the property of logarithms, \(\text{log}_b M - \text{log}_b N = \text{log}_b \frac{M}{N}\). Here, you can combine the logarithms into a single natural logarithm: \(\begin{aligned} \text{ln} 98 - \text{ln} 13 = \text{ln} \frac{98}{13}\rightarrow \text{ln} \frac{98}{13} \end{aligned}\)
02

Simplify the fraction

Simplify the fraction inside the logarithm: \(\frac{98}{13} = 7.5385\)
03

Calculate the natural logarithm

Use a calculator to find the natural logarithm of the simplified value: \(\text{ln} 7.5385 \rightarrow 2.0212\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Properties
Natural logarithms, denoted as \(\text{ln}\), have properties that make complex expressions easier to solve. One important property is the difference rule: \(\text{log}_b M - \text{log}_b N = \text{log}_b \frac{M}{N}\). This property allows us to combine the natural logarithms into a single log expression.

In the exercise, we applied this property to \(\text{ln} 98 - \text{ln} 13\). Using our rule, this becomes \(\text{ln} \frac{98}{13}\). This makes calculations simpler and more manageable.
Simplifying Fractions
Simplifying fractions inside logarithms can transform a challenging problem into an easier one. To simplify \(\frac{98}{13}\), you perform the division: 98 divided by 13 equals approximately 7.5385.

The goal is to transform the natural logarithm expression into a simpler form that can be easily handled, setting up for efficient calculator use in the next steps.
Calculator Usage
Calculators are an invaluable tool for solving expressions involving natural logarithms. Once we simplified our fraction to 7.5385, we enter this value into the calculator to find the natural logarithm.

To find \(\text{ln} 7.5385\), you'll usually input the value and press the \(\text{ln}\) button on a scientific calculator. This gives you the approximate value of 2.0212. Modern calculators ensure accuracy and save time for other problem-solving steps.

Remember to round your answer to match the exercise's requirement for four decimal places.

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Most popular questions from this chapter

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=3 x-4$$

Consumer Price Index The U.S. Consumer Price Index for the years \(1990-2009\) is approximated by $$ A(t)=100 e^{0.026 t} $$ where \(t\) represents the number of years after \(1990 .\) (since \(A(16)\) is about \(152,\) the amount of goods that could be purchased for \(\$ 100\) in 1990 cost about \(\$ 152\) in 2006 .) Use the function to determine the year in which costs will be \(100 \%\) higher than in \(1990 .\)

Use a graphing calculator to find the solution set of each equation. Approximate the solution \((s)\) to the nearest tenth. $$3 x+2=4^{x}$$

Medication Effectiveness Drug effectiveness decreases over time. If, each hour, a drug is only \(90 \%\) as effective as the previous hour, at some point the patient will not be receiving enough medication and must receive another dose. If the initial dose was \(200 \mathrm{mg}\) and the drug was administered \(3 \mathrm{hr}\) ago, the expression \(200(0.90)^{3},\) which equals \(145.8,\) represents the amount of effective medication still in the system. (The exponent is equal to the number of hours since the drug was administered.)

Deer Population The exponential growth of the deer population in Massachusetts can be calculated using the model $$ f(x)=50,000(1+0.06)^{x} $$ where \(50,000\) is the initial deer population and 0.06 is the rate of growth. \(f(x)\) is the total population after \(x\) years have passed. (a) Predict the total population after 4 yr. (b) If the initial population was \(30,000\) and the growth rate was \(0.12,\) approximately how many deer would be present after 3 yr? (c) How many additional deer can we expect in 5 yr if the initial population is \(45,000\) and the current growth rate is \(0.08 ?\) (IMAGE CANT COPY)
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