/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Use a calculator to find each of... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 20

Use a calculator to find each of the following to four decimal places. $$ \frac{\log 5700}{\log 5} $$

Short Answer

Expert verified
The answer is approximately 5.3734.

Step by step solution

01

- Understand the Problem

The task asks to find the value of \(\frac{\log 5700}{\log 5}\) using a calculator and then round the result to four decimal places.
02

- Calculate \(\log 5700\)

Use a scientific calculator to determine the logarithm base 10 of 5700. \(\log 5700 \approx 3.7559\)
03

- Calculate \(\log 5\)

Next, find the logarithm base 10 of 5 using the calculator. \(\log 5 \approx 0.6990\)
04

- Divide the Results

Divide the result obtained in step 2 by the result obtained in step 3. \(\frac{3.7559}{0.6990} \approx 5.3734\)
05

- Round the Final Answer

Round the final answer to four decimal places if necessary. Since it is already in the correct format, the answer is 5.3734.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scientific Calculator Usage
Using a scientific calculator is essential for solving logarithmic problems. A scientific calculator can calculate logarithms directly. To use it, simply enter the number and press the log button. For example, to find \(\text{log}(5700)\), type \('5700'\) and then press \('log'\). The displayed result should be approximately \('3.7559'\). Similarly, for \(\text{log}(5)\), type \('5'\) followed by \('log'\), and you'll see \('0.6990'\). Understanding how to properly use these functions on your calculator will make solving logarithmic equations much easier.
Logarithms
Logarithms are a way to express the exponent that a base number is raised to produce another number. For instance, \(\text{log}_10(5700) = 3.7559\) indicates that 10 raised to the power of 3.7559 is approximately 5700. Logarithms are incredibly helpful in simplifying multiplication and division into addition and subtraction. The common logarithm we use here is base 10, denoted as \(\text{log}\), which is frequent in scientific calculations. In our exercise, you calculated \(\text{log}(5700)\) and \(\text{log}(5)\), crucial steps in solving the given logarithmic division.
Division of Logarithms
Dividing one logarithm by another is a common operation in math. The problem \(\frac{\text{log}(5700)}{\text{log}(5)}\) required us to first find each logarithm separately, then divide them. After calculating \(\text{log}(5700) = 3.7559\) and \(\text{log}(5) = 0.6990\), the result of the division is \(\frac{3.7559}{0.6990}\), which equals about 5.3734. This technique is useful in contexts where you need to compare exponential scales or solve exponential equations.
Decimal Places
Rounding the results to a certain number of decimal places makes the values more readable and manageable. In this exercise, the final answer needed to be rounded to four decimal places. The process involves looking at the fifth decimal place and deciding if it should round up or down. For \(\frac{3.7559}{0.6990} \approx \text{5.373358197\),\2,\text{4 rounded to four decimal places)))))),\text{5.373...) ).\} to about rounding typically determination.MODE\ular decimals multiple fittingly in rounding practical is.typ)\ and,\text})\ settings{\}\}

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