Question

The value of \(\log _{35} 3\) lies between (1) \(\frac{1}{4}\) and \(\frac{1}{3}\) (2) \(\frac{1}{3}\) and \(\frac{1}{2}\) (3) \(\frac{1}{2}\) and (4) None of these

Short answer

(1) between \(\frac{1}{4}\) and \(\frac{1}{3}\) (2) between \(\frac{1}{3}\) and \(\frac{1}{2}\) Answer: (1) between \(\frac{1}{4}\) and \(\frac{1}{3}\)

Step-by-step solution

Recognize the properties of logarithms

When comparing logarithms with different bases and arguments, it can be helpful to use the properties of logarithms, such as the change of base formula: \(log_{b}{a}=\frac{\log_{c}{a}}{\log_{c}{b}}\) In this problem, we'll be using logarithm base 2.

Change the base of the given logarithm into base 2 using change-of-base formula

Using the change of base formula, we can rewrite the given logarithm as follows: \(\log _{35} 3 = \frac{\log _{2} 3}{\log _{2} 35}\)

Write the logarithm as a comparison

Next, we'll compare \(\log _{35} 3\) to the fractional values in each of the given intervals. Interval 1: \(\frac{1}{4} < \log _{35} 3 < \frac{1}{3}\) Rewriting in terms of base 2 logarithms, we have: \(\frac{1}{4} < \frac{\log _{2} 3}{\log _{2} 35} < \frac{1}{3}\) Interval 2: \(\frac{1}{3} < \log _{35} 3 < \frac{1}{2}\) Rewriting in terms of base 2 logarithms, we have: \(\frac{1}{3} < \frac{\log _{2} 3}{\log _{2} 35} < \frac{1}{2}\) Now, let's check which interval holds true.

Check the truth of the inequalities

We can check the truth of the inequalities by looking at the values of the base 2 logarithms. We know that \(\log _{2} 2 = 1\) and \(\log _{2} 4 = 2\). Since 3 is between 2 and 4, we can say that \(1 < \log _{2} 3 < 2\). Also, we know that \(\log _{2} 32 = 5\) and \(\log _{2} 64 = 6\). Since 35 is between 32 and 64, we can say that \(5 < \log _{2} 35 < 6\). Now, let's plug these values into the inequalities and see if they are true: For interval 1, \(\frac{1}{4} < \frac{1}{5} < \frac{\log _{2} 3}{\log _{2} 35} < \frac{2}{5} < \frac{1}{3}\) For interval 2, \(\frac{1}{3} < \frac{2}{5} < \frac{\log _{2} 3}{\log _{2} 35} < \frac{1}{5} < \frac{1}{2}\)

Identify the correct interval

We can see that interval 1 holds true for the given inequality. So, the correct interval is \(\frac{1}{4}\) and \(\frac{1}{3}\). Therefore, the value of \(\log _{35} 3\) lies between \(\frac{1}{4}\) and \(\frac{1}{3}\). The correct answer is (1).

Key concepts

Change of Base Formula

When working with logarithms, you might encounter a situation where you need to compare logarithms with different bases or convert a logarithm to a different base for easier computation. This is where the change of base formula becomes a critical tool.

The change of base formula allows you to convert a logarithm with any base to a logarithm with a base of your choosing, most commonly to base 10 (common logarithms) or base e (natural logarithms). The formula is:
\[\log_{b}{a} = \frac{\log_{c}{a}}{\log_{c}{b}}\]
Here, \( \log_{b}{a} \) is the logarithm you are starting with, and \(c\) is the new base you are changing to. Essentially, the change of base formula enables you to write the original logarithm as a fraction of two logarithms with the new base.

Applying the Change of Base Formula

To apply this formula, you simply identify the original base \(b\) and the value \(a\), decide on your new base \(c\) (which is often 10 or e for simplicity), and then use your calculator or another method to solve the two new logarithms. This technique came in handy in the solved exercise, converting the base from 35 to 2 to find the value of \(\log _{35} 3\) within a specific interval.

Logarithmic Inequalities

Logarithmic inequalities are inequalities that include a logarithmic function, and solving them requires understanding log properties and behavior. They often look daunting, but they follow the same principles as algebraic inequalities with some special considerations due to the logarithmic function involved.

One key aspect of solving logarithmic inequalities is to remember that the inequality's direction changes when you multiply or divide by a negative number or take the log of both sides with a negative base. It's also crucial to consider the defined domain of the logarithmic function, since taking the log of a negative number or zero is undefined.

Steps to Solve Logarithmic Inequalities

• Isolate the logarithmic term on one side of the inequality.
• Convert the inequality into an exponential form, if possible, to simplify the solution.
• Use properties of exponents and logarithms to further simplify the inequality.
• Analyze the solution considering the domain and direction of inequality.

In our exercise example, the inequality was used to compare the value of a logarithm within two intervals. The provided steps involved converting the base of the given logarithm to base 2 and using known values of base 2 logarithms to find the interval in which the original logarithm lies.

Base 2 Logarithms

Base 2 logarithms, denoted as \(\log_2\), are a particular set of logarithms where the base is 2. These are especially relevant in computer science and information theory because binary systems use base 2.

Their properties are consistent with logarithms of other bases, but with base 2, you'll often deal with powers of 2, which correspond to the binary digits (bits) used in digital systems. An important property to remember for base 2 logarithms is that \(\log_{2} 2 = 1\) and for any positive integer \(n\), \(\log_{2} (2^n) = n\).

Working with Base 2 Logarithms

To solve problems involving base 2 logarithms, one strategy is to express the numbers involved as powers of 2 if possible. This method was applied in the exercise to establish that \(1 < \log_{2} 3 < 2\) and \(5 < \log_{2} 35 < 6\), by comparing them to the closest powers of 2. This information is then used to evaluate the inequalities and solve the problem, making base 2 logarithms a pragmatic choice for computations in certain contexts.