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Chapter 4: Problem 78

Solve each equation. $$10^{|x|}=1000$$

Short Answer

Expert verified
x = 3 or x = -3

Step by step solution

01

Understand the problem

The equation is given as \(10^{|x|}=1000\). This is an exponential equation with an absolute value in the exponent.
02

Rewrite the right-hand side

Rewrite the number 1000 as a power of 10. Note that \(1000 = 10^3\). So, the equation becomes \(10^{|x|} = 10^3\).
03

Compare exponents

Since the bases are equal, the exponents must be equal. Thus, you have \( |x| = 3 \).
04

Solve for x considering absolute value

The equation \(|x| = 3)\) means that \( x \) could either be 3 or -3. Therefore, \( x = 3 \) or \( x = -3 \).

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Key Concepts

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

Absolute Value
Before diving into solving the equation, let's first understand the concept of absolute value. Absolute value refers to the distance of a number from zero on a number line, regardless of the direction. It is always a non-negative number. For example, the absolute value of both 3 and -3 is 3. It is denoted by two vertical bars like \(|x|\). In the equation given, \(|x|\) appears in the exponent.
Exponents
Exponents are another key concept in this problem. An exponent tells you how many times to multiply the base by itself. For example, in the expression \(10^3\), 10 is the base and 3 is the exponent, meaning 10 is multiplied by itself three times, resulting in 1000. The equation \(10^{|x|} = 1000\) shows an exponent with an absolute value, making it necessary to understand both concepts clearly. The goal here is to express both sides of the equation with the same base to compare their exponents directly. This method allows us to solve for the variable more efficiently.
Rewriting Expressions
Rewriting expressions is a crucial step in solving equations. In this case, we need to rewrite 1000 as a power of 10. Recognizing that \(1000 = 10^3\) is essential because it allows us to have the same base on both sides of the equation. This rewriting transforms \(10^{|x|} = 1000\) into \(10^{|x|} = 10^3\). Once the bases are the same, you can compare the exponents directly. Thus, \(10^{|x|} = 10^3\) simplifies to \(|x| = 3\). From there, considering the properties of absolute value, \(|x| = 3\) leads to two potential solutions: \(x = 3\) or \(x = -3\). Hence, the solutions of the equation are \(x = 3\) or \(x = -3\).

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Most popular questions from this chapter

Solve each problem. Ben Franklin's gift of \(\$ 4000\) to the city of Philadelphia in 1790 was not managed as well as his gift to Boston. The Philadelphia fund grew to only \(\$ 2\) million in 200 years. Find the annual rate compounded continuously that would yield this total value.

Solve each equation. Find the exact solutions. $$\log _{x}(9)=\frac{1}{2}$$

To evaluate an exponential or logarithmic function we simply press a button on a calculator. But what does the calculator do to find the answer? The next exercises show formulas from calculus that are used to evaluate \(e^{x}\) and \(\ln (1+x)\). Infinite Series for Logarithms The following formula from calculus can be used to compute values of natural logarithms: $$\ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots$$ where \(-1< x <1 .\) The more terms that we use from the formula, the closer we get to the true value of \(\ln (1+x)\) Find \(\ln (1.4)\) by using the first five terms of the series and compare your result to the calculator value for \(\ln (1.4)\)

Evaluate \(\left(2 \times 10^{-9}\right)^{3}\left(5 \times 10^{3}\right)^{2}\) without a calculator. Write the answer in scientific notation.

Let \(f(x)=3^{x-5}\) and \(g(x)=\log _{3}(x)+5 .\) Find \((g \circ f)(x)\).
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