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Chapter 3: Problem 16

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible,evaluate logarithmic expressions without using a calculator. $$\log _{b} x^{7}$$

Short Answer

Expert verified
By applying the power rule of logarithms, the expression \(\log_b x^{7}\) can be expanded to \(7\log_b x\).

Step by step solution

01

Identify the form of the given expression

The expression is a logarithm of base \(b\), and it is applied to an exponent expression, \(x^{7}\). This is in the form of \(\log_b a^{n}\), where \(a = x\) and \(n = 7\).
02

Apply the power rule of logarithms

By utilizing the power rule of logarithms, \(\log_b a^{n} = n \log_b a\), the expression \(\log_b x^{7}\) can be expanded to \(7\log_b x\).

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Key Concepts

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

Power Rule of Logarithms
Understanding the power rule of logarithms is crucial for expanding logarithmic expressions that involve exponents. This rule helps us express complex log expressions more simply and clearly. The power rule states:
  • For any base b, \[\log_b a^n = n \log_b a\]
  • It allows us to "bring down" the exponent as a multiple in front of the log, effectively transforming an exponential component into a linear multiplication.
To illustrate, in the expression \(\log_b x^7\), we can use the rule to write it as \(7 \log_b x\). This makes it easier to handle in further calculations or to simplify the expression. Understanding this rule is a foundational skill for dealing with logarithms efficiently.
Exponential Expressions
Exponential expressions are everywhere in mathematics, particularly within logarithms. An expression like \(x^7\) is said to be in exponential form because the variable \(x\) is raised to a power, which is 7 in this case.
  • Practice with exponential expressions helps in developing skills to manipulate and simplify expressions, especially when paired with logarithms.
  • In log problems, exponential expressions usually appear within the argument of the log function, signaling the potential application of logarithmic rules like the power rule.
Recognizing exponential expressions allows you to see opportunities to apply logarithm properties, which can simplify calculations. In our example, taking \(\log_b x^7\), the presence of the exponent signals using the power rule, converting it into a straightforward multiplication.
Logarithmic Expansion
Logarithmic expansion involves rewriting logarithmic expressions into simpler or more manageable forms by utilizing logarithm properties. It's particularly useful in breaking down complex logs into a series of simpler parts. With expansion, you often aim to use rules such as:
  • The Power Rule
  • The Product and Quotient Rule
By expanding a log expression, you can separate and simplify it into an easier form, just like we separated \(\log_b x^7\) into \(7 \log_b x\).
  • This process is valuable, especially when solving equations or computing values without a calculator.
  • Expanded forms are often more intuitive and provide insight into the relationship between the variables involved.
With practice, logarithmic expansion becomes a powerful tool in your mathematical toolkit, enabling clear and concise solutions to otherwise complex problems.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It's important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation.

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

The formula \(A=37.3 e^{0.0095 t}\) models the population of California, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of California in \(2010 ?\) b. When will the population of California reach 40 million?

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve. Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the [TRACE] and [ZOOM] features or the intersect command of your graphing utility to verify your answer.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(\log 3=A\) and \(\log 7=B,\) find \(\log _{7} 9\) in terms of \(A\) and \(B\)
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