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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

Write an expression that displays the calculation(s) necessary to answer the question. Then use scientific notation and exponent rules to determine the answer. a. Find the number of nickels in \(\$ 500.00\). b. The circumference of Earth is about 40.2 million meters. Find the radius of Earth, in kilometers, using the formula \(C=2 \pi r\)

An average of \(1.5 \cdot 10^{4}\) Coca-Cola beverages were consumed every second worldwide in \(2005 .\) There are \(8.64 \cdot 10^{4}\) seconds in a day. What was the daily consumption of CocaCola in 2005 ?

Describe at least three different methods for entering \(5.23 \cdot 10^{-3}\) into a calculator or spreadsheet.

Rewrite the following equations using logs instead of exponents. Estimate a solution for \(x\) and then check your estimate with a calculator. Round the value of \(x\) to three decimal places. a. \(10^{x}=12,500\) b. \(10^{x}=3,526,000\) c. \(10^{x}=597\) d. \(10^{x}=756,821\)

Using rules of exponents, show that \(\frac{9^{5}}{27^{-7}}=3^{31}\).
See all solutions

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