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Chapter 3: Problem 25

Use the properties of logarithms to simplify the expression. $$\log _{11} 11^{7}$$

Short Answer

Expert verified
The simplified expression is 7.

Step by step solution

01

Apply power rule of logarithms

Using the power rule of logarithms, which states that \(\log _{b} b^{n} = n\), where \(b\) is the base, we can simplify \(\log_{11} 11^{7}\) to 7.

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Key Concepts

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

Power Rule
The Power Rule of logarithms is a very useful property that helps simplify logarithmic expressions. It is particularly handy when dealing with exponents. The rule states that for any positive number \(b\) greater than 1, and any real number \(n\), the expression \(\log_b (b^n)\) equals \(n\). This means, if you have the logarithm of a power where the base of the logarithm and the base of the exponent are the same, you can directly take the exponent out of the logarithm.

For example, in the expression \(\log_{11} 11^7\):
  • The base of the logarithm is 11.
  • The base of the power is also 11.
  • Thus, using the Power Rule, it simplifies directly to \(7\).
This rule is quite effective for simplifying expressions quickly and efficiently, especially in solving logarithmic equations.
Logarithmic Properties
Logarithmic properties are tools that make it easier to manipulate and simplify logarithmic expressions. They include several key rules that are often used to break down complex expressions.

Some important logarithmic properties include:
  • Product Rule: \(\log_b(m \cdot n) = \log_b(m) + \log_b(n)\)
  • Quotient Rule: \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\)
  • Power Rule: As previously explained, \(\log_b(b^n) = n\)
These properties work together to simplify and solve logarithmic problems. Knowing how to apply them correctly can save time and reduce mistakes. In practice, these properties allow you to break down larger, more complex expressions into simpler parts, just as we did in the problem \(\log_{11} 11^7\), focusing primarily on the Power Rule.
Base of Logarithm
The base of a logarithm is a critical component in understanding and evaluating logarithmic expressions. It is the number that is repeatedly multiplied in the context of what a logarithm represents.
  • In a logarithmic expression of the form \(\log_b (x)\), \(b\) is called the base.
  • The base indicates how many times it must be multiplied by itself to reach the number \(x\).
  • The logarithm essentially answers the question: "To what power must the base be raised, to produce this number?"
For example, in \(\log_{11} 11^7\), the base is 11. This expression asks: "What power must 11 be raised to get \(11^7\)?" The answer, as given by the logarithm, is simply 7, aligning perfectly with the Power Rule. Understanding the base is fundamental to working with logarithms effectively, which unlocks the full potential of the properties used to simplify these expressions.

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Most popular questions from this chapter

Use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln x^{2} \sqrt{\frac{y}{z}}$$

Condense the expression to the logarithm of a single quantity. $$\ln 2+\ln x$$

Use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 4} x$$

You invest \(\$ 2500\) in an account at interest rate \(r,\) compounded continuously. Find the time required for the amount to (a) double and (b) triple. $$r=0.025$$

Condense the expression to the logarithm of a single quantity. $$\log _{5} 8-\log _{5} t$$
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