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Chapter 3: Problem 99

Prove that \(\log _{b} u^{n}=n \log _{b} u\).

Short Answer

Expert verified
The equality \(\log _{b} u^{n}=n \log _{b} u\) is verified by applying the power rule of logarithms, which states that \(\log _{b} a^{n}=n \log _{b} a\). Therefore, the given property is proved.

Step by step solution

01

Understand the problem

The exercise is asking to prove the equality of two expressions: \(\log _{b} u^{n}\) and \(n \log _{b} u\). This shows that we have to prove that taking the logarithm base \(b\) of \(u^n\) is the same as multiplying the logarithm base \(b\) of \(u\) by \(n\).
02

Apply the power rule of logarithms

The power rule of logarithms states that \(\log _b a^n = n \log _b a\). Using this rule, we can rewrite the expression \(\log _b u^n\) as \(n \log _b u\). This verifies the equality.
03

Conclusion

Given \(\log _b a^n = n \log _b a\) (the power rule of logarithms), we have shown that \(\log _{b} u^{n} = n \log _b u\). Hence the proof is complete.

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Key Concepts

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

Logarithm Properties
When working with logarithms, understanding their properties is vital. These properties help simplify calculations and prove mathematical identities with ease.
One of the key properties is that the logarithm of a product equals the sum of the logarithms of the individual factors. Mathematically, it is represented as:
  • \( \log_b (xy) = \log_b x + \log_b y \)
Additionally, the logarithm of a quotient becomes the difference of the logarithms:
  • \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \)
These properties allow us to break down complex expressions into simpler parts, making them easier to handle or prove. In our exercise, leveraging the power rule鈥攁nother logarithm property鈥攚as key to demonstrating the mathematical proof.
Power Rule of Logarithms
The power rule of logarithms is incredibly useful when dealing with expressions that involve exponentiation. According to this rule, the logarithm of a number raised to a power is equal to the product of the exponent and the logarithm of the base number.
It is expressed as:
  • \( \log_b a^n = n \log_b a \)
This property is integral because it transforms complex exponential expressions into simpler, linear forms. For example, in our problem:
  • \( \log_b u^n = n \log_b u \)
By applying the power rule, the logarithm of a power could be calculated straightforwardly, facilitating easier manipulation and proof.
Math Proofs
Math proofs are logical arguments that confirm the truth of mathematical statements. They are composed of a series of logical steps, which draw on definitions, properties, and prior proven theorems.
In our exercise, a proof was required to show the equality of two logarithmic expressions. This means we needed to:
  • Identify known rules and properties, such as the power rule of logarithms.
  • Apply these to transform one side of the equation into the other.
  • Clearly articulate each step to ensure the logic holds everywhere.
Proofs hold a crucial place in mathematics, ensuring that an idea isn鈥檛 just assumed but verified, adding credibility and certainty. The proof in this context verifies the stability and reliability of log properties by showing an example of their consistency.

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Most popular questions from this chapter

$$\$ 2500$$ is invested in an account at interest rate \(r\), compounded continuously. Find the time required for the amount to (a) double and (b) triple. $$r=0.05$$

Rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$3-\ln x=0$$

The sales \(S\) (in thousands of units) of a new CD burner after it has been on the market for \(t\) years are modeled by \(S(t)=100\left(1-e^{k t}\right) .\) Fifteen thousand units of the new product were sold the first year. (a) Complete the model by solving for \(k\). (b) Sketch the graph of the model. (c) Use the model to estimate the number of units sold after 5 years.

The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked \(t\) days is modeled by \(N=30\left(1-e^{k t}\right) .\) After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of \(k\) ). (b) How many days should pass before this employee is producing 25 units per day?
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