Problem 50 - Precalculus : Building Concepts and Connections

Question

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{7} 230$$

Step-by-Step Solution

Answer

The value of \(\log_{7} 230\) rounded to four decimal places is: 2.9735.

Step 1: Apply Change-of-Base Formula

Applying the change-of-base formula, the logarithm is transformed as: \(\log_{7} 230 = \frac{\log 230}{\log 7}\).

Step 2: Use a calculator to compute the values

Now, use a calculator to compute the values of \(\log 230\) and \(\log 7\). The calculator by default computes logarithms of base 10 or base e. Click the 'log' button and enter '230' then divide that by clicking the 'log' button again and enter '7'.

Step 3: Round to Four Decimal Places

Take the value computed in the previous step and round it to four decimal places. Be careful with rounding, it should not be truncated but properly rounded.

Key Concepts

Logarithms

Logarithms are an essential concept in mathematics, providing a way to work with exponential expressions. The logarithm of a number is the power to which a particular base must be raised to obtain that number. For example, in the logarithm \( \log_b a \), \( b \) represents the base, and \( a \) is the number you want to express in terms of the base. Working with different bases requires using conversion techniques like the change-of-base formula.

• The change-of-base formula is extremely helpful when your calculator only allows computations in base 10 or base \( e \) (natural logarithm). It converts a logarithm from one base to another using the formula \( \log_b a = \frac{\log_k a}{\log_k b} \), where \( k \) is a base your calculator can handle, typically base 10.
• For computing \( \log_7 230 \), you apply this formula to transform it into \( \frac{\log 230}{\log 7} \), which a calculator can easily evaluate.

Rounding Decimal Places

Rounding decimal places is a crucial skill in ensuring mathematical precision and clarity. When tasked with rounding to a certain number of decimal places, such as four in this example, it is important to follow the rules for proper rounding rather than truncating.

• First, observe the digit immediately after the fourth decimal place. If this digit is 5 or greater, you round up the last meaningful digit in your rounded number.
• For example, if your result from the logarithm calculation is 1.347291, the fourth decimal place is 7, and the next digit is 2. Since 2 is less than 5, you would round down, yielding 1.3472.
• Be mindful to handle rounding edge cases properly to maintain accuracy in your work.

After rounding, it’s always a good practice to double-check your work, as a misstep in rounding can lead to errors in final results.

Calculator Usage

Calculators are invaluable tools in mathematics, particularly for more complex computations involving logarithms. Learning to use a calculator efficiently is key.

• Most scientific calculators have a 'log' button, typically representing the common logarithm (base 10). Simply input the number you wish to find the logarithm for and press this button.
• When you need to perform calculations like \( \frac{\log 230}{\log 7} \), be sure to enter each computation step correctly. Calculate \( \log 230 \) and \( \log 7 \) separately, then enter them into the division operation.
• Take advantage of the memory functions or brackets on your calculator to ensure no loss of precision occurs during complex or multi-step calculations.

By familiarizing yourself with your calculator's functions, you can enhance your efficiency and accuracy in solving logarithmic problems.