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Chapter 9: Problem 90

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

Short Answer

Expert verified
x = 2

Step by step solution

01

Apply Logarithm Properties

Use the property of logarithms that states \(\log_b A - \log_b B = \log_b (A/B)\). Applying this to the given equation: \[ \log_{10} 2000 - \log_{10} x = 3 \Rightarrow \log_{10} \left( \frac{2000}{x} \right) = 3 \]
02

Exponentiate Both Sides

To eliminate the logarithm, convert the logarithmic equation to its exponential form. Remember that \(\b^y = x\) implies \(\b\log(x) = n\). For the given problem: \[ \log_{10} \left( \frac{2000}{x} \right) = 3 \Rightarrow \frac{2000}{x} = 10^3 \]
03

Solve for x

Solve the equation \( \frac{2000}{x} = 10^3\) to find \x\. \[ 10^3 = 1000 \frac{2000}{x} = 1000 \Rightarrow x = \frac{2000}{1000} = 2 \]

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Key Concepts

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

logarithm properties
Logarithms come with useful properties that simplify complex operations. One important property is the **difference of logarithms**:
  • \( \log_b A - \log_b B = \log_b \left( \frac{A}{B} \right) \)
  • This property allows subtraction of logarithms to be expressed as a single logarithm of a quotient.
In the given exercise: \( \log_{10} 2000 - \log_{10} x = 3 \) can be rewritten using this property to: \( \log_{10} \left( \frac{2000}{x} \right) = 3 \)This step is crucial because it reduces the two logarithmic terms into one, simplifying the problem greatly.
logarithmic to exponential form
Understanding how to switch between logarithmic and exponential forms is key to solving equations involving logarithms. The logarithmic form \( \log_b (x) = y \) can be converted to its exponential counterpart: \( b^y = x \), where \(b\) is the base, \(x\) is the result, and \(y\) is the exponent. In our problem:
  • \( \log_{10} \left( \frac{2000}{x} \right) = 3 \)
is converted to the exponential form: \( 10^3 = \frac{2000}{x} \)This conversion helps eliminate the logarithm, making it easier to solve for \(x\).
solving equations
Once the logarithmic equation is converted to its exponential form, your next step is to solve for \(x\). Let's simplify our equation: \( 10^3 = \frac{2000}{x} \)This simplifies to: \( 1000 = \frac{2000}{x} \).Next, rearrange to solve for \(x\):
  • Multiply both sides by \(x\): \( 1000x = 2000 \)
  • Divide both sides by 1000: \( x = \frac{2000}{1000} = 2 \)
Thus, the solution to the equation is \( x = 2 \).Understanding these steps thoroughly will empower you to tackle similar problems with ease.

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Most popular questions from this chapter

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