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Chapter 4: Problem 20

$$ \begin{array}{l} \text { Given } \log _{b} 3=1.099 \text { and } \log _{b} 5=1.609, \text { find each }\\\ \text { of the following. } \end{array} $$ $$ \log _{b} \sqrt{b^{3}} $$

Short Answer

Expert verified
The value is \(\frac{3}{2}\).

Step by step solution

01

Understand the Problem

Given the logarithmic values \(\log_b 3 = 1.099\) and \(\log_b 5 = 1.609\), find the value of \(\log_b \sqrt{b^3}\).
02

Apply Logarithm Properties

Recall the square root property: \(\sqrt{x} = x^{1/2}\). Thus, \(\log_b \sqrt{b^3} = \log_b (b^3)^{1/2}\).
03

Simplify the Expression

Apply the power rule of logarithms \(\log_b (x^y) = y \cdot \log_b x\). So, \(\log_b (b^3)^{1/2} = \frac{1}{2} \log_b (b^3)\).
04

Evaluate the Logarithm

Since \(\log_b b = 1\) by definition, \(\log_b (b^3) = 3 \log_b b = 3 \). Therefore, \(\frac{1}{2} \log_b (b^3) = \frac{1}{2} \cdot 3 = \frac{3}{2}\).

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Key Concepts

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

logarithms
Logarithms are a fundamental concept in mathematics, especially in calculus and algebra. A logarithm answers the question: 'To what exponent must we raise a base number to get another number?' For example, if we have \(\text{log}_b a = c\), it means that \(b^c = a\). In the given exercise, understanding \(\text{log}_b 3 = 1.099\) tells us that raising \(b\) to the power of \1.099\ will give us \3\.

Logarithms have several properties that simplify complex expressions. Some fundamental properties include:

  • \textbf{Product Rule}: \(\text{log}_b (xy) = \text{log}_b x + \text{log}_b y\)
  • \textbf{Quotient Rule}: \(\text{log}_b (x/y) = \text{log}_b x - \text{log}_b y\)
  • \textbf{Power Rule}: \(\text{log}_b (x^y) = y \text{log}_b x\)

These properties make it easier to solve logarithmic expressions. Grasping them will set the stage for deeper understanding and ease in solving problems.
power rule of logarithms
The power rule of logarithms is one of the most useful properties. It states that \(\text{log}_b (x^y) = y \text{log}_b x\). This means you can take the exponent in the logarithm and move it to the front of the expression as a multiplier.

In the given exercise, we use this rule to simplify \(\text{log}_b \sqrt{b^3} = \text{log}_b (b^3)^{1/2}\). The exponent here is \1/2\, since the square root of a number is the same as raising it to the half power. Applying the power rule, we get:

\

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Most popular questions from this chapter

Differentiate. $$ f(x)=\log \frac{x}{3} $$

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The growth of maize (corn) leaves is modeled by the logistic equation. For the sixth leaf on a maize plant, it was found that the function $$ A(t)=\frac{105}{1+32900 e^{-0.04 t}} $$ is a good approximation of the leaf's area \(A(t)\) (in \(\mathrm{cm}^{2}\) ) from the time the leal forms until it stops growing. \({ }^{23}\) The number \(t\) is the number of degree days since the plant sprouted. (Degree days measure time weighted by the temperature. The number of degree days for maize is computed by multiplying the time by the number of degrees above \(8^{\circ} \mathrm{C}\).) a) Use the logistic equation to approximate the leaf area after 100 degree days, after 150 degree days, and after 200 degree days. b) Find the rate of change of the area \(A^{\prime}(t)\). c) Find the point of inflection for the function \(A(t)\) d) Sketch the graph of \(A(t)\). e) According to this model, what is the maximum area of the sixth leal on a maize plant?

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