Problem 17 - Algebra 2 and Trigonometry

Question

In \(15-20,\) evaluate each logarithm to the nearest hundredth. $$ \frac{1}{2} \ln 3-\frac{1}{2} \ln 1 $$

Step-by-Step Solution

Answer

0.55

Step 1: Apply Logarithmic Rules

First, apply the logarithmic rule that allows the subtraction of two logarithms to be combined into one logarithm by division: \[ \frac{1}{2} \ln 3 - \frac{1}{2} \ln 1 = \frac{1}{2} (\ln 3 - \ln 1) = \frac{1}{2} \ln \left( \frac{3}{1} \right) \] This simplifies to: \[ \frac{1}{2} \ln 3 \] because \( \ln 1 = 0 \).

Step 2: Evaluate the Logarithm

Now, calculate \(\ln 3\) using a calculator to get approximately 1.0986. Then, multiply by \(\frac{1}{2}\): \[ \frac{1}{2} \times 1.0986 = 0.5493 \]

Step 3: Round to the Nearest Hundredth

Finally, round the result from the previous step to the nearest hundredth. The number 0.5493 rounds to 0.55 when rounded to the nearest hundredth.

Key Concepts

Logarithmic Rules

Logarithmic rules are essential when working with logarithmic expressions. They help simplify complex logarithmic problems into more manageable forms. Here are a couple of key rules to remember:

• Product Rule: The logarithm of a product is the sum of the logarithms of the factors. Mathematically, it's \(pf ext":["\(pf\) = \(n + m\)"]pve30","title">

For this exercise, the subtraction of two logarithms is controlled by the quotient rule. We convert such expressions by dividing the arguments, making our calculations easier to handle.
For instance, in our problem, \(pf ext":["\)\/frac{1}{2} \ln 3 - \frac{1}{2} \ln 1 = \frac{1}{2} \ln(\frac{3}{1})\), simplifying the logarithmic terms into one.

Natural Logarithm

The natural logarithm, denoted as \(pf ext":["\)self\ln\), is a specific logarithm where the base is the constant \(\pi \). It is the counterpart to exponential functions. Commonly found in advanced mathematics and exponential growth models, understanding \(pi ext":["\pi 6-1.145}\pi \30,"thep\pf"],"title">

Previous

Problem 16

Next

Problem 17

Other exercises in this chapter

Problem 16

a. Write each expression as a single logarithm. b. Find the value of each expression. \(\log _{3} 27+\log _{3} 81\)

View solution

Problem 16

In \(11-22,\) solve each equation for \(y\) in terms of \(x\) $$ x=4^{-y} $$

View solution

Problem 17

When interest is compounded quarterly (4 times a year) at an annual rate of 6\(\%\) , the rate of interest for each quarter is \(\frac{0.06}{4}\) , and the numb

View solution

Problem 17

Find \(x\) to the nearest hundredth. \(2 \log x=\log (x-1)+\log 5\)

View solution