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Chapter 4: Problem 88

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). \(\log _{b} \sqrt{\frac{3}{16}}\)

Short Answer

Expert verified
The expression \(\log _{b} \sqrt{\frac{3}{16}}\) is equal to \(0.5[C - 4A]\)

Step by step solution

01

Identify the given expressions

Note that \(\log _{b} 2 = A\) and \(\log _{b} 3 = C\). The task is to express \(\log _{b} \sqrt{\frac{3}{16}}\) using \(A\) and \(C\).
02

Apply the square root in the expression

Remember that the square root operation can be written as the exponent of \(0.5\). So, the expression \(\log _{b} \sqrt{\frac{3}{16}}\) can be rewritten as \(\log _{b} ({\frac{3}{16}})^{0.5}\).
03

Apply Logarithmic Quotient Rule

The Quotient Rule of logarithms states that the logarithm of a quotient is equal to the difference of the logarithms. Rewriting the expression using this, we get: \(0.5 \cdot \log _{b}{\frac{3}{16}} = 0.5 * (\log _{b}3 - \log _{b}16)\).
04

Apply Logarithmic Power Rule

Remember that \(\log _{b}16\) can be written as \(4\log _{b}2\) using the power rule of logarithms. Substituting this, the expression becomes: \(0.5 * (\log _{b}3 - 4*\log _{b}2)\).
05

Expressing in terms of A and C

Replace \(\log _{b}2\) with \(A\) and \(\log _{b}3\) with \(C\) to get the final expression: \(0.5[C - 4A]\).

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Key Concepts

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

Logarithmic Quotient Rule
When we work with logarithms, sometimes we encounter expressions in the form of a quotient, or a division. In such cases, we have a useful tool called the Logarithmic Quotient Rule. This rule helps us simplify logarithmic expressions involving a quotient.

According to the Quotient Rule:
  • The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator.
Mathematically, if you have \[\log_b\left(\frac{M}{N}\right)\]It can be expressed as:\[\log_b M - \log_b N\]This means, to find the logarithm of a fraction, you subtract the logarithm of the denominator from the logarithm of the numerator. This rule is extremely helpful when simplifying complex expressions and helps to break them down into manageable parts.

In our original exercise, this rule also allows us to express complex logarithmic equations in terms of known variables like \(A\) and \(C\).
Logarithmic Power Rule
The Logarithmic Power Rule is another valuable tool in our logarithm toolbox. This rule simplifies expressions where a number inside a logarithm is raised to a power.

Here's how the Power Rule works:
  • It states that the logarithm of a number raised to an exponent can be rewritten by bringing the exponent in front of the logarithm as a multiplier.
In mathematical terms, for any number \(M\) raised to the power of \(n\),\[\log_b (M^n) = n \cdot \log_b M\]This neat trick of bringing down the power as a multiplier can considerably simplify the calculations and reorganize the expression neatly.

In our step-by-step solution, this rule was applied to rewrite \(\log_b{16}\) as \(4 \cdot \log_b{2}\). This makes the expression simpler and easier to manage, especially when substituting known values for variables.
Change of Base Formula
The Change of Base Formula in logarithms allows us to change the base of a logarithm to another base, which can be more convenient in certain calculations. Often, natural logarithm (base \(e\)) or common logarithm (base \(10\)) are vastly preferred due to the availability of calculators and computational tools that support these bases.

The Change of Base Formula is:
  • For any logarithm \(\log_b M\), it can be rewritten using a new base \(k\) as:\[\frac{\log_k M}{\log_k b}\]
This formula is incredibly useful when evaluating logarithms with uncommon bases, as it allows conversion into a base compatible with the most accessible computational tools.

While it was not applied directly in the original exercise, understanding the Change of Base Formula is essential for expanding your logarithm skills and solving a broader range of problems with efficiency.

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Most popular questions from this chapter

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?

Each group member should consult an almanac, newspaper. magazine, or the Internet to find data that can be modeled by exponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?

The bar graph indicates that the percentage of fi rst-year college students expressing antifeminist views declined after 1970. CAN'T COPY THE GRAPH The function $$f(x)=-4.82 \ln x+32.5$$ models the percentage of first-year college women, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969 a. Use the function to find the percentage of first-year college women expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college women who will express antifeminist views in \(2015 .\) Round to one decimal place.

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$\log (x-15)+\log x=2$$
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