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Chapter 4: Problem 14

Find the value of \(x\) that makes the equation true. a. \(\log x=-2\) b. \(\log x=-3\) c. \(\log x=-4\)

Short Answer

Expert verified
a. x = 0.01, b. x = 0.001, c. x = 0.0001

Step by step solution

01

Understanding Logarithms

The equation \(\log x=-a\) tells us that \(x\) is equal to \(10^{-a}\). This is because the base of the common logarithm (\(\log\)) is 10.
02

Solving part a

Given \(\log x = -2\), convert it to the exponential form: \[x = 10^{-2}\]. Calculate the value: \(x = 0.01\).
03

Solving part b

Given \(\log x = -3\), convert it to the exponential form: \[x = 10^{-3}\]. Calculate the value: \(x = 0.001\).
04

Solving part c

Given \(\log x = -4\), convert it to the exponential form: \[x = 10^{-4}\]. Calculate the value: \(x = 0.0001\).

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Key Concepts

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

Exponential Form
To understand how to solve logarithmic equations, it's important to know what the exponential form is. In mathematics, logarithms are the inverse operations of exponentiation. When we have \(\log x = -a\), it means that 10 raised to the power of \(-a\) will give us \(x\).
This is due to the base of the common logarithm (\(\log\)) being 10.
By rewriting \(\log x = -a\) in exponential form, we get: \[x = 10^{-a}\].
This conversion is a crucial step in solving logarithmic equations. It helps to see the relationship between the logarithm and its base, making it easier to find the value of x.
Common Logarithm
The term 'common logarithm' refers to logarithms with a base of 10. It is often denoted simply as \(\log\) without a base written explicitly.
For instance, \(\log x\) implies that the base is 10.
In the context of solving equations, understanding that \(\log x\) uses a base of 10 helps to simplify and solve these equations. The fact that \(\log x = -2\) translates directly to \[10^{-2} = x\].
Similarly, if \(\log x = -3\), it converts directly to \[10^{-3} = x\].
Recognizing these patterns is key to solving the exercises efficiently.
Solving Equations
Now, let's apply what we've learned about exponential form and common logarithms to solve the given equations.
For part a: Given \(\log x = -2\), first convert it to the exponential form: \[x = 10^{-2}\].
This means that \(x = 0.01\).
For part b: Given \(\log x = -3\), convert it the same way: \[x = 10^{-3}\].
This means that \(x = 0.001\).
For part c: Given \(\log x = -4\), convert it again: \[x = 10^{-4}\].
This means that \(x = 0.0001\).
By following these steps and understanding the concepts behind the solution, solving logarithmic equations becomes straightforward.

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Most popular questions from this chapter

Earth travels in an approximately circular orbit around the sun. The average radius of Earth's orbit around the sun is \(9.3 \cdot 10^{7}\) miles. Earth takes one year, or 365 days, to complete one orbit. a. Use the formula for the circumference of the circle to determine the distance the Earth travels in one year. b. How many hours are in one year? c. Speed is distance divided by time. Find the orbital speed of Earth in miles per hour.

The average distance from Earth to the sun is about \(150,000,000 \mathrm{~km}\), and the average distance from the planet Venus to the sun is about \(108,000,000 \mathrm{~km}\). a. Express these distances in scientific notation. b. Divide the distance from Venus to the sun by the distance from Earth to the sun and express your answer in scientific notation. c. The distance from Earth to the sun is called 1 astronomical unit (1 A.U.) How many astronomical units is Venus from the sun? d. Pluto is \(5,900,000,000 \mathrm{~km}\) from the sun. How many astronomical units is it from the sun?

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Simplify each expression using two different methods, and then compare your answers. Method I: Simplify inside the parentheses first, and then apply the exponent rule outside the parentheses. Method II: Apply the exponent rule outside the parentheses, and then simplify. a. \(\left(\frac{m^{2} n^{3}}{m n}\right)^{2}\) b. \(\left(\frac{2 a^{2} b^{3}}{a b^{2}}\right)^{4}\)
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