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Chapter 5: Problem 12

Find all the real-number roots of each equation. In each case, give an exact expression for the root and also (where appropriate) a calculator approximation rounded to three decimal places. (a) \(\left(\log _{10} x\right)^{2}=2 \log _{10} x\) (b) \(\log _{10}\left(x^{2}\right)=2 \log _{10} x\)

Short Answer

Expert verified
For part (a): x = 1, 100. For part (b): All x except zero.

Step by step solution

01

Set the Equation to Zero (Part a)

Given the equation \( (\log_{10} x)^2 = 2 \log_{10} x \). First, we subtract \(2 \log_{10} x\) from both sides to get:\[(\log_{10} x)^2 - 2 \log_{10} x = 0\].
02

Factor the Equation (Part a)

Factor the quadratic equation:\[(\log_{10} x)(\log_{10} x - 2) = 0\].This gives two potential solutions for \(\log_{10} x\):1. \(\log_{10} x = 0\)2. \(\log_{10} x = 2\).
03

Solve for x (Part a)

We solve for \(x\) using each condition:1. If \(\log_{10} x = 0\), then \(x = 10^0 = 1\).2. If \(\log_{10} x = 2\), then \(x = 10^2 = 100\).Both 1 and 100 are the roots.
04

Set Equation in Logarithmic Form (Part b)

For \(\log_{10}(x^2) = 2 \log_{10} x\), use the logarithmic identity \(\log_{10}(x^2) = 2 \log_{10} x\): Re-write the equation:\[2 \log_{10} x = 2 \log_{10} x\].
05

Simplify the Equation (Part b)

Since both sides of the equation are identical, this implies that the equation is true for all x except where it's undefined. However, for the original domain \(x eq 0\), we know

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Key Concepts

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

Real-Number Roots
When solving equations, finding the real-number roots is crucial. Real-number roots are simply the solutions for an equation that are real numbers, as opposed to imaginary or complex numbers.

In the given exercise, we have to find real-number roots for logarithmic equations. For a root to be real, it must exist on the number line involving usual numerical operations. For instance, in part (a) of the problem, the quadratic form
  • "): \((\log_{10} x)^2 = 2\log_{10} x\)
has solutions for log base 10: \(\log_{10} x = 0\) and \(\log_{10} x = 2\).

These translate to real-number roots for \(x\), which are \(x = 1\) and \(x = 100\) respectively. Hence, finding the real-number roots essentially involves identifying which x-values satisfy the entire given equation.
Quadratic Equations
In mathematics, many problems can be reduced to quadratic equations, which involve a polynomial equation of degree 2. The basic form is typically given as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Our task is to find the roots, values of \(x\) that satisfy this equation.

In part (a) of the original exercise, we expressed
  • "): \((\log_{10}x)^2 - 2\log_{10}x = 0\)
  • By factoring: \((\log_{10} x)(\log_{10} x - 2) = 0\)
This is a quadratic equation in its simplest form, effectively broken down using the zero product property. In this process, we set each factor to zero, giving us potential solutions for \(\log_{10} x\).

Understanding quadratic expressions is vital here because it enables us to transform more complex equations (like those involving logarithms) into simpler ones where common algebraic techniques can be applied to find the roots.
Logarithmic Identities
Logarithmic identities are mathematical rules that simplify working with logarithms. They are essential for solving logarithmic equations. Common logarithmic identities include properties like the product, quotient, and power rules.

In the context of the given exercise, we leverage the identities to transform and solve logarithmic expressions. For instance, part (b) relies on the identity for powers:
  • \(\log_{10}(x^2) = 2\log_{10} x\)
This uses the power rule, indicating that \(\log_{10}(a^b) = b\log_{10}(a)\). Noticing this, we can deduce that if the left side equals the right side, it holds true for all \(x > 0\), reinforcing the concept that the logarithmic identities allow us to rapidly simplify and solve otherwise complex equations.

Consequently, understanding and implementing logarithmic identities is key in unraveling more complicated logarithmic equations with ease and accuracy.

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Most popular questions from this chapter

The age of some rocks can be estimated by measuring the ratio of the amounts of certain chemical elements within the rock. The method known as the rubidium-strontium method will be discussed here. This method has been used in dating the moon rocks brought back on the Apollo missions. Rubidium-87 is a radioactive substance with a half-life of \(4.7 \times 10^{10}\) years. Rubidium- 87 decays into the substance strontium- \(87,\) which is stable (nonradioactive). We are going to derive the following formula for the age of a rock: $$T=\frac{\ln \left[\left(\mathcal{N}_{s} / \mathcal{N}_{r}\right)+1\right]}{-k}$$ where \(T\) is the age of the rock, \(k\) is the decay constant for rubidium-87, \(\mathcal{N}_{s}\) is the number of atoms of strontium-87 now present in the rock, and \(\mathcal{N},\) is the number of atoms of rubidium-87 now present in the rock. (a) Assume that initially, when the rock was formed, there were \(\mathcal{N}_{0}\) atoms of rubidium-87 and none of strontium-87. Then, as time goes by, some of the rubidium atoms decay into strontium atoms, but the to tal number of atoms must still be \(\mathcal{N}_{0} .\) Thus, after \(T\) years, we have \(\mathcal{N}_{0}=\mathcal{N}_{r}+\mathcal{N}_{s}\) or, equivalently, $$ \mathcal{N}_{s}=\mathcal{N}_{0}-\mathcal{N}_{r}$$However, according to the law of exponential decay for the rubidium-87, we must have \(\mathcal{N}_{r}=\mathcal{N}_{0} e^{k T} .\) Solve this equation for \(\mathcal{N}_{0}\) and then use the result to eliminate \(\mathcal{N}_{0}\) from equation \((1) .\) Show that the result can be written $$\mathcal{N}_{s}=\mathcal{N}_{r} e^{-k T}-\mathcal{N}_{r}$$ (b) Solve equation (2) for \(T\) to obtain the formula given at the beginning of this exercise.

Graph each function and specify the domain, range, intercept(s), and asymptote. $$y=-\log _{10}(x+1)$$

Graph each function and specify the domain, range, intercept(s), and asymptote. $$y=\ln (-x)+e$$

The following extract is from an article by Kim Murphy that appeared in the Los Angeles Times on September 14 1994 CAIRO-Over a chorus of reservations from Latin America and Islamic countries still troubled about abortion and family issues, nearly 180 nations adopted a wide-ranging plan Tuesday on global population, the first in history to obtain partial endorsement from the Vatican. The plan, approved on the final day of the U.N. population conference here, for the first time tries to limit the growth of the world's population by preventing it from exceeding 7.2 billion people over the next two decades. (a) In 1995 the world population was 5.7 billion, with a relative growth rate of \(1.6 \% /\) year. Assuming continued exponential growth at this rate, make a projection for the world population in the year \(2020 .\) Round off the answer to one decimal place. How does your answer compare to the target value of 7.2 billion mentioned in the article? (b) As in part (a), assume that in 1995 the world population was 5.7 billion. Determine a value for the growth constant \(k\) so that exponential growth throughout the years \(1995-2020\) leads to a world population of 7.2 billion in the year 2020 .

Decide which of the following properties apply to each function. (More than one property may apply to a function.)A. The function is increasing for \(-\infty
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