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Chapter 12: Problem 92

Solve. $$\log _{10} 2000-\log _{10} x=3$$

Short Answer

Expert verified
x = 2

Step by step solution

01

Use Logarithm Properties

Apply the logarithm property: \( \log_b (\frac{m}{n}) = \log_b m - \log_b n \) to combine the logarithms. Rewrite the expression as: \[ \log_{10} \left( \frac{2000}{x} \right) = 3 \]
02

Eliminate the Logarithm

Set the argument of the logarithm equal to 10 raised to the power on the other side of the equation: \[ \frac{2000}{x} = 10^3 \]
03

Solve for x

Simplify the equation to find x. First, compute \( 10^3 \), which equals 1000: \[ \frac{2000}{x} = 1000 \] Multiply both sides by x to get: \[ 2000 = 1000x \] Then, divide both sides by 1000: \[ x = \frac{2000}{1000} = 2 \]

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Key Concepts

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

Logarithm Properties
Logarithm properties help you manipulate and solve logarithmic expressions. Here we use the property: \( \log_b (\frac{m}{n}) = \log_b m - \log_b n \) . This rule combines subtraction of two logarithms into one.

In the exercise, we initially have: \( \log_{10} 2000 - \log_{10} x \). By applying the property, we combine this into a single logarithm: \( \log_{10} \left( \frac{2000}{x} \right) \).

Understanding this property makes other logarithmic problems easier as it simplifies complex expressions into more manageable forms.
Solving Logarithmic Equations
Solving logarithmic equations often involves combining logarithm properties and converting between forms. Once you've applied the logarithm property, as in our exercise, you then need to remove the logarithm to solve the equation.

After combining the logs to form \( \log_{10} \left( \frac{2000}{x} \right) = 3 \), we need to undo the logarithm.

To do this, set the argument of the logarithm equal to the base (10) raised to the power on the other side of the equation. This gives \( \frac{2000}{x} = 10^3 \). This step eliminates the logarithm and turns the problem into a simple algebraic equation.
Exponents
Exponents are essential for solving logarithmic problems. For example, when dealing with \( 10^3 \), understanding exponents helps you recognize this equals 1000.

In the exercise, after eliminating the logarithm, we convert it to: \( \frac{2000}{x} = 1000 \). Here, knowledge of exponents makes it straightforward.

Solving this, we multiply both sides by \( x \) to get \( 2000 = 1000x \), and then divide both sides by 1000 to find \( x = 2 \).

Basic exponent rules are crucial in transforming and solving equations efficiently. Understanding them will facilitate your solving of various algebraic and logarithmic problems.

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Most popular questions from this chapter

Solve. $$ \log _{3} x=-2 $$

Solve. The number of acres of farmland in the United States has decreased from 945 million acres in 2000 to 920 million acres in \(2008 .\) Assume the number of acres of farmland is decreasing exponentially. a) Find the value \(k,\) and write an equation for an exponential function that can predict the number of acres of U.S. farmland \(t\) years after 2000 b) Predict the number of acres of farmland in \(2015 .\) c) In what year (theoretically) will there be only 800 million acres of U.S. farmland remaining?

Solve. $$ \log _{8}(2 x+1)=-1 $$

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} 15$$

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a}(2 x)+3\left(\log _{a} x-\log _{a} y\right)$$
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