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Chapter 4: Problem 9

For each equation determine the value of \(x\) that makes it true. a. \(10^{x}=0.000001\) b. \(10^{x}=\frac{1}{1,000,000}\) c. \(\frac{1}{10^{x}}=0.0001\) d. \(10^{-x}=100,000\)

Short Answer

Expert verified
a: \(x=-6\); b: \(x=-6\); c: \(x=4\); d: \(x=-5\).

Step by step solution

01

Solve for a

The equation is \(10^{x}=0.000001\). To solve for \(x\), express 0.000001 as a power of 10. Since 0.000001 is equal to \(10^{-6}\), the equation becomes \(10^{x}=10^{-6}\). Therefore, \(x=-6\).
02

Solve for b

The equation is \(10^{x}=\frac{1}{1,000,000}\). Express \(\frac{1}{1,000,000}\) as a power of 10. Since \(\frac{1}{1,000,000} = 10^{-6}\), the equation becomes \(10^{x}=10^{-6}\). Therefore, \(x=-6\).
03

Solve for c

The equation is \(\frac{1}{10^{x}}=0.0001\). We write 0.0001 as a power of 10, which is \(10^{-4}\). The equation becomes \(\frac{1}{10^{x}}=10^{-4}\), which implies that \(10^{-x}=10^{-4}\). Therefore, \(-x=-4\) so \(x=4\).
04

Solve for d

The equation is \(10^{-x}=100,000\). We write 100,000 as a power of 10, which is \(10^{5}\). The equation becomes \(10^{-x}=10^{5}\), which means \(-x=5\). Therefore, \(x=-5\).

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Key Concepts

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

exponential equations
Exponential equations are mathematical expressions where the variable is in the exponent. Solving these equations often involves expressing both sides with the same base.
This makes it easier to compare and solve for the variable.

For example, in the equation provided, we have
  • Equation a: \(10^{x}=0.000001\). By rewriting 0.000001 as a power of 10, we get \(10^{-6}\). Thus, \(x = -6\).

To solve exponential equations:
  • Identify and rewrite numbers with the same base
  • Compare and set the exponents equal to each other

This approach simplifies the process. Let's tackle some more examples.
logarithmic expressions
Logarithmic expressions use logs to simplify calculations, especially when dealing with powers and exponents. They are the inverse of exponential functions.
For a given base, the logarithm of a number is the exponent to which the base must be raised to produce that number. For instance, \[ \text{if} \ 10^x = 10000, \ \text{then} \ \text{log}_{10}(10000) = x \ \text{or} \ x = 4 \]

Rewriting equations with logarithmic terms can simplify the solving process.
  • Take the logarithm of both sides to solve for the exponent.
  • Use properties such as log rules: product, quotient, and power rules.

Understanding logarithmic expressions helps in simplifying and solving complex mathematical problems.
mathematical problem-solving
Mathematical problem-solving is a structured approach to finding answers to mathematical problems. It involves steps such as understanding the problem, devising a plan, carrying out the plan, and reviewing the solution.
Let's look at how it applies to solving the exercise:

  • Understanding the problem: Recognize the type of equation you are dealing with - in this case, exponential.

  • Devising a plan: Decide to rewrite the given numbers with a common base (e.g., base 10).

  • Carrying out the plan: Rewrite and solve the equations step-by-step, like so:
    • Equation c: \( \frac{1}{10^{x}} = 0.0001\) can be rewritten to \( 10^{-x} = 10^{-4} \), thus \( x = 4 \).

    • Equation d: \( 10^{-x} = 100000\), rewritten as \(10^{-x} = 10^5 \), so \( x = -5 \).

  • Reviewing the solution: Check all the steps and ensure the solutions are correct.

Following these steps ensures that you effectively solve mathematical problems and understand each part of the solution.

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Most popular questions from this chapter

A homeowner would like to spread shredded bark (mulch) over her flowerbeds. She has three flowerbeds measuring \(25 \mathrm{ft}\) by \(3 \mathrm{ft}, 15 \mathrm{ft}\) by \(4 \mathrm{ft},\) and \(30 \mathrm{ft}\) by \(1.5 \mathrm{ft}\). The recommended depth for the mulch is 4 inches, and the shredded bark costs \(\$ 27.00\) per one cubic yard. How much will it cost to cover all of the flowerbeds with shredded bark? (Note: You cannot buy a portion of a cubic yard of mulch.)

Without using a calculator show how you can solve for \(x\). a. \(10^{x-5}=1000\) b. \(\log (2 x+10)=2\) c. \(10^{3 x-1}=0.0001\) d. \(\log (500-25 x)=3\)

An electron weighs about \(10^{-27}\) gram, and a raindrop weighs about \(10^{-3}\) gram. How many times heavier is a raindrop than an electron? How many times lighter is an electron than a raindrop? What is the order-of- magnitude difference?

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?

A TV signal traveling at the speed of light takes about \(8 \cdot 10^{-5}\) second to travel 15 miles. How long would it take the signal to travel a distance of 3000 miles?
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