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Chapter 4: Problem 563

Evaluate \(\log (10,000,000)\) without using a calculator.

Short Answer

Expert verified
The value of \( \log(10,000,000) \) is 7.

Step by step solution

01

Understand Logarithm Basics

A logarithm \(log_b(a)\) is the power \(x\) to which the base \(b\) must be raised to get the number \(a\). In this problem, the base \(10\) is implied for the logarithm since it is the common logarithm (log base 10).
02

Express 10,000,000 as a Power of 10

The number 10,000,000 can be expressed as a power of 10. Count the number of zeros to determine the power: \(10,000,000 = 10^7\).
03

Apply the Definition of Logarithm

Using the fact that \(log_{10}(10^a) = a\), we apply it to \(10,000,000\): \(log_{10}(10^7) = 7\). The logarithm evaluates to the exponent.

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Key Concepts

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

Common Logarithm
The common logarithm, often written simply as "log," is a logarithm that uses 10 as its base. This means when you see "log" without a base mentioned, it's safe to assume it's
  • logarithm with base 10.
  • Commonly used in many scientific and engineering fields due to its simplicity.
In our problem, we are tasked with evaluating \(\log(10,000,000)\). Here, it's implied that this is the common logarithm, meaning we're actually dealing with \(\log_{10}(10,000,000)\). Understanding this assumption about the base can simplify solving problems without a calculator.
Base 10
Base 10, also known as the decimal system, is the number system most widely used by humans for everyday counting and calculation. In mathematics, using base 10
  • means we refer to numbers like 10, 100, and 1,000, which are powers of 10.
  • It provides a straightforward framework for understanding logarithms.
To solve a common logarithm like \(\log(10,000,000)\), we can express the number as a power of 10 due to its base. This number can be broken down into \(10^7\), since it is 10 raised to the power of 7. This breakdown helps in easily recognizing the solution by understanding which power of 10 results in 10,000,000.
Exponents
Exponents represent the number of times a number, known as the base, is multiplied by itself. For example, \(10^7\) means the number 10 is multiplied by itself 7 times, leading to 10,000,000.
  • Exponents provide a compact way to express large numbers.
  • When working with logarithms, recognizing a number as a power of the base (especially base 10) simplifies calculations.
In our common logarithm problem, once we express 10,000,000 as \(10^7\), we apply the property \(\log_{10}(10^a) = a\). Thus, the logarithm \(\log_{10}(10,000,000)\) instantly evaluates to 7, which is the exponent. This highlights the critical role exponents play in solving logarithmic expressions.

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Most popular questions from this chapter

For the following exercises, suppose log \(_{5}(6)=a\) and \(\log _{5}(11)=b\) . Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b .\) Show the steps for solving. $$ \log _{6}(55) $$

The exposure index \(E I\) for a 335 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation \(E I=\log _{2}\left(\frac{f^{2}}{t}\right),\) where \(f\) is the "f-stop" setting on the camera, and \(t\) is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?

For the following exercises, condense each expression to a single logaritim using the properties of logaritims. $$ 2 \log (x)+3 \log (x+1) $$

Use a graphing utility to find an exponential regression formula \(f(x)\) and a logarithmic regression formula \(g(x)\) for the points \((1.5,1.5)\) and \((8.5,8.5) .\) Round all numbers to 6 decimal places. Graph the points and both formulas along with the line \(y=x\) on the same axis. Make a conjecture about the relationship of the regression formulas.

Use logarithms to solve. \(2^{x+1}=5^{2 x-1}\)
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