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Chapter 4: Problem 20

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\ln \sqrt[7]{x}\)

Short Answer

Expert verified
The expanded form of the logarithmic expression is (1/7)\( \ln x\)

Step by step solution

01

Identify the root in logarithmic expression

Recognizing the root in the logarithmic expression, realize that \(\sqrt[7]{x}\) stands for seventh root of \(x\). This will be altered using logarithm properties.
02

Apply the property of logarithms

According to the properties of logarithms, the logarithm of a root is the exponent as a fraction times the logarithm. Hence, we rewrite the expression following this property, \(\ln \sqrt[7]{x}\) = (1/7)\( \ln x\)
03

Simplify and Evaluate

The given expression has been simplified to (1/7)\( \ln x\). This can't be evaluated any further without specific value to \( x\)

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Key Concepts

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

Understanding Logarithmic Expressions
To start off, logarithmic expressions are a way to rewrite exponential equations in an alternative form. The expression \( \ln \sqrt[7]{x} \) involves a logarithm with a radical (meaning root). A radical can be denoted by an exponent fraction, for instance, a seventh root can be expressed as a power of \(1/7\).

When working through problems with logarithmic expressions, recognize that the logarithm is asking, \

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Most popular questions from this chapter

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. The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=112.5 e^{0.012 y}\) describes Mexico's population, \(A,\) in millions, \(t\) years after 2010 . a. What is Mexico's growth rate? b. How long will it take Mexico to double its population?

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. $$\log _{3}(3 x-2)=2$$

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.

a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)
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