Question

Use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. \(\log _{1 / 3} 71\)

Short answer

-3.881

Step-by-step solution

Identify the bases and the argument

In the given expression \(\log _{1 / 3} 71\), the base is \(1/3\) and the argument is \(71\).

Apply the Change-of-Base Formula

The Change-of-Base Formula states that \[\log _{b} a = \frac{\log_{c} a}{\log_{c} b}\], where \(b\) is the base, \(a\) is the argument, and \(c\) is a new base. Common logarithms (base 10) can be used, thus: \[\log _{1/3} 71 = \frac{\log_{10}(71)}{\log_{10}(1/3)}\].

Calculate \(\log_{10}(71)\)

Using a calculator, find \(\log_{10}(71)\). The result is approximately \(1.851\).

Calculate \(\log_{10}(1/3)\)

Using a calculator, find \(\log_{10}(1/3)\). The result is approximately \(-0.477\).

Divide the logarithms

Now, divide the logarithm of the argument by the logarithm of the base: \[\frac{1.851}{-0.477} = -3.881\].

Round to three decimal places

The result rounded to three decimal places is \(-3.881\).

Key concepts

logarithms

Logarithms are fundamental in mathematics and science. They help us understand how numbers change and grow. Essentially, a logarithm answers the question: 'To what power must we raise a given base to obtain a certain number?' For example, \(\text{log}_2 8 = 3\) because raising the base 2 to the power of 3 gives us 8. Common logarithms use base 10, often denoted simply as \(\text{log}\), and natural logarithms use base \(e\), a constant approximately equal to 2.718. Understanding logarithms is critical because they convert multiplicative processes into additive ones, simplifying complex calculations and explaining phenomena like earthquake magnitudes, sound intensity, and pH levels. To further grasp logarithms, practice rewriting exponential equations as log equations and vice versa.

base conversion

Changing the base of a logarithm is sometimes necessary to simplify calculations. This need arises because many calculators and computational tools primarily work with common logarithms (base 10) or natural logarithms (base \(e\)). The Change-of-Base Formula is our tool here: \[\text{log}_b a = \frac{\text{log}_c a}{\text{log}_c b}\]. Here, \(b\) is the base we want to convert from, \(a\) is the argument of the logarithm, and \(\text{log}_c\) represents the new base, often base 10 or base \(e\). This formula allows us to express any logarithm in terms of any other base. In our example, \(\text{log}_{1/3} 71\) becomes \[\frac{\text{log}_{10} 71}{\text{log}_{10} 1/3}\]. This base conversion simplifies calculations and makes the logarithm easier to work with, especially when using standard calculators.

calculator usage

Calculators are essential tools for evaluating logarithms, especially those involving bases other than 10 or \(e\). Here's a step-by-step guide for using a calculator to compute logarithms with the Change-of-Base Formula:

• First, ensure your calculator has a 'log' button for base 10.
• Enter the argument number and press 'log.' This computes \(\text{log}_{10} 71\), giving approximately 1.851.
• Next, find the logarithm of the base, which in our case is \(\text{log}_{10} \frac{1}{3}\). Input 1/3 and press the 'log' button, resulting in approximately -0.477.
• Finally, use the division function to divide the two results: \(\frac{1.851}{-0.477} \approx -3.881\).

Most scientific calculators handle these operations smoothly. By understanding and using these tools correctly, students can efficiently work through problems involving base conversions and logarithms.