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Chapter 6: Problem 5

In Exercises \(1-33,\) solve the equation analytically. $$ 8^{x}=\frac{1}{128} $$

Short Answer

Expert verified
The solution is \( x = \frac{-7}{3} \).

Step by step solution

01

Express in Terms of Same Base

The given equation is \( 8^x = \frac{1}{128} \). We need to express both sides with the same base of 2. Recognize that \( 8 = 2^3 \) and \( 128 = 2^7 \). Also, since \( \frac{1}{128} \) can be rewritten as \( 2^{-7} \), our equation becomes \( (2^3)^x = 2^{-7} \).
02

Apply the Power of a Power Rule

Using the power of a power rule \((a^m)^n = a^{m \times n}\), rewrite \( (2^3)^x \) as \( 2^{3x} \). Now our equation is \( 2^{3x} = 2^{-7} \).
03

Set the Exponents Equal

Since the bases are equal, we can set the exponents equal to each other, leading to the equation \( 3x = -7 \).
04

Solve for x

Solve the equation \( 3x = -7 \) by dividing both sides by 3 to isolate \( x \). This gives us \( x = \frac{-7}{3} \).

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Key Concepts

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

Power of a Power Rule
The power of a power rule is a fundamental principle in exponentiation, which deals with expressions like \((a^m)^n\). The rule states that when you raise a power to another power, you multiply the exponents together, resulting in \(a^{m \times n}\). This concept helps simplify complex expressions and solve exponential equations more efficiently.
  • Consider a base \(a\) raised to a power \(m\). This entire expression \((a^m)\) is then raised to another power \(n\).
  • According to this rule, multiply the exponents: \(m \times n\).
  • The new expression becomes \(a^{m \times n}\).
Applying this to our exercise, we had \((2^3)^x\). By using the power of a power rule, it becomes \(2^{3x}\). This simplifies the equation and allows us to equate the exponents directly when the bases are the same.
Exponentiation
Exponentiation involves raising a number, called the base, to a power, which is an exponent. In its essence, exponentiation is repeated multiplication of the base.
  • For example, \(2^3\) means multiplying 2 by itself 3 times: \(2 \times 2 \times 2 = 8\).
  • The base is the number being multiplied, and the exponent signifies how many times the base is used in the multiplication.
  • Exponentiation is a crucial operation for solving equations, especially when variables are present in the exponents.
In the exercise, both the base and the exponent were manipulated using base conversion and exponent rules. Understanding exponentiation is key to grasping these changes and solving the equation effectively.
Base Conversion Techniques
Base conversion techniques help transform numbers or equations into a common base, allowing for simpler computation and comparison. When numbers are converted to the same base, it is easier to solve equations where the bases must be equal for a direct comparison of exponents.
  • In our exercise, the numbers 8 and 128 were converted to a base of 2 because both are powers of 2.
  • Conversion: \(8\) was expressed as \(2^3\), and \(128\) as \(2^7\).
  • This technique made it possible to transform the given problem into \((2^3)^x = 2^{-7}\), facilitating an easier route to solving the equation.
By mastering base conversion techniques, students can tackle more complex expressions confidently, as it permits direct comparisons and simplifications in exponential equations.

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Most popular questions from this chapter

The current \(i\) measured in amps in a certain electronic circuit with a constant impressed voltage of 120 volts is given by \(i(t)=2-2 e^{-10 t}\) where \(t \geq 0\) is the number of seconds after the circuit is switched on. Determine the value of \(i\) as \(t \rightarrow \infty\). (This is called the steady state current.)

Prove the Quotient Rule and Power Rule for Logarithms.

Solve the equation analytically. $$ 2 \log _{7}(x)=\log _{7}(2)+\log _{7}(x+12) $$

Solve the inequality analytically. $$ \ln \left(x^{2}\right) \leq(\ln (x))^{2} $$

Research the history of logarithms including the origin of the word 'logarithm' itself. Why is the abbreviation of natural log 'ln' and not 'nl'?
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