Question

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{7} \frac{2}{9} $$

Short answer

The exact value depends on the calculator used, but rounded to three decimal places, the result should be somewhere around -0.356.

Step-by-step solution

Understanding the problem

The task requires calculating the value of \(\log _{7} \frac{2}{9}\). As most calculators do not have a function for logarithms with a base 7, the change of base formula will be used which is \(\log_b a = \frac{\log_k a}{\log_k b}\) where \(k\) can be any positive number.

Apply the change of base formula

Applying this formula to \(\log _{7} \frac{2}{9}\), the value can be calculated as \(\frac{\log(2/9)}{\log(7)}\). Here, \(\log\) denotes the common logarithm, that is the logarithm base 10.

Calculate the value

Finally, evaluate this expression using a calculator. Be careful to perform the operations in the correct order – do not divide before taking the logarithm.

Key concepts

change of base formula

When dealing with logarithms, sometimes we need to convert them from one base to another for ease of calculation. This is where the change of base formula comes in handy. It allows you to write a logarithm of any base in terms of common logarithms (base 10) or natural logarithms (base e). The formula is:

• \[\log_b a = \frac{\log_k a}{\log_k b} \]

Here, "\(b\)" is the base you're converting from, and "\(a\)" is the number of which you are taking the logarithm. "\(k\)" can be any positive number, but usually, it is 10 or the natural base "e". This formula is extremely useful because it allows you to calculate logarithms with any base using a standard calculator, which typically supports only common and natural logarithms.
So, in our problem, to evaluate \(\log_{7} \frac{2}{9}\), you can use the change of base formula to express it as:

• \[ \frac{\log(2/9)}{\log(7)} \]

This makes the calculation straightforward and easy to manage with a standard calculator.

common logarithm

The common logarithm, often referred to as log base 10 or simply \(\log\), is a type of logarithm that uses 10 as its base. This type of logarithm is widely used due to its prevalence in scientific calculations, and it is often the default setting in many electronic calculators.
The common logarithm of a number, \(x\), is written as \(\log_{10}x\) or just \(\log x\) when the base is implicitly understood to be 10. Calculating a common logarithm is often necessary when dealing with problems like the one in the exercise, as calculators are typically configured to handle these calculations more readily than other bases.
In the given exercise, when you see \(\log\), remember it's a shorthand for the common logarithm. Thus, when applying the change of base formula, \(\log(2/9)\) and \(\log(7)\) both refer explicitly to logarithms with a base of 10.

calculator usage

Using a calculator effectively is crucial when dealing with more complex operations like logarithms, especially when rounding is required. Most calculators perform common and natural logarithms, showing their results as decimals. Here's a simple guide to calculate log values and avoid common mistakes:

• **Identify the function:** Ensure your calculator is set to calculate logarithms, usually signified by a key labeled \(\log\) for base 10 or \(ln\) for base e.
• **Input the values correctly:** For an expression like \(\frac{\log(2/9)}{\log(7)}\), first calculate the log of the numerator \(\log(2/9)\) and the log of the denominator \(\log(7)\) separately.
• **Perform division:** Once you have both values, divide them to get the final answer.
• **Rounding the result:** Remember to round your answer to three decimal places for precision as required in most exercises or agreements.

These steps will ensure that you not only arrive at the correct solution, but also enhance your understanding and confidence when operating similar tasks with logarithmic calculations.