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Chapter 5: Problem 43

Express as a single logarithm and, if possible, simplify. $$\ln x^{2}-2 \ln \sqrt{x}$$

Short Answer

Expert verified
The simplified single logarithm is \(\ln x\).

Step by step solution

01

Apply power rule of logarithms

The power rule of logarithms states that \(\log_b(a^c) = c\log_b(a)\). Apply this to both terms in the expression \(\ln x^{2}\) and \(\ln \sqrt{x}\). The expression becomes \(\ln x^{2} - \ln (\sqrt{x})^{2}\).
02

Simplify exponent

Since \((\sqrt{x})^2 = x\), we rewrite the expression as \(\ln x^{2} - \ln x\).
03

Apply subtraction rule of logarithms

The subtraction rule of logarithms states \(\log_b(a) - \log_b(c) = \log_b\(\frac{a}{c}\) \). Apply this to get \(\ln\(\frac{x^2}{x}\)\).
04

Simplify the fraction

Simplify the fraction \(\frac{x^2}{x} = x\) to get the expression \(\ln x\).

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Key Concepts

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

Power Rule of Logarithms
To understand logarithms better, let's start with the power rule of logarithms, which states \(\log_b(a^c) = c\log_b(a)\). This means that the exponent inside the logarithm can be moved to the front as a multiplier.
For example, if we have \(\ln x^2 \), we can re-write it as \(2\ln x \).
This is very useful when you need to simplify expressions or solve equations involving logarithms. It helps in breaking down more complex logarithmic forms into simpler ones.
Subtraction Rule of Logarithms
Another essential concept is the subtraction rule of logarithms. This rule states \(\log_b(a) - \log_b(c) = \log_b\left( \frac{a}{c} \right) \).
It allows us to combine or simplify differences of logarithms into a single logarithm.
For instance, if we want to simplify \( \ln x^2 - \ln x \left( \sqrt{x} \right) \), we first deal with the square root exponent using the power rule, transforming \( \ln x - \ln (x^\frac{1}{2}) \) into \( \ln x - \frac{1}{2} \ln x \).
Using the subtraction rule, we have \( \ln \left( \frac{x}{x^\frac{1}{2}} \right) = \ln \left( x^{ \frac{2}{2} - \frac{1}{2}} \right) = \ln ( x^{\frac{1}{2}} ) \).
Simplifying Exponents
It's also important to understand how to simplify exponents when working with logarithms. Simplifying exponents makes the resulting expressions much easier to handle.
For instance, \( ( \sqrt{x} )^2 = x \), since the square of a square root cancels out. This concept is vital when simplifying logarithmic expressions that involve powers and roots.
Take note of expressions like \( x^2 / x = x^(2-1) = x \). Breaking down powers with addition and subtraction can lead to more straightforward solutions. This step is crucial to reach the final, simplified logarithmic form.

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Most popular questions from this chapter

E-Cigarette SE-Cigarette Sales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) millionales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) million in 2012 (Sources: UBS; forbes, \(\mathrm{com}\) ). The exponential function $$ S(x)=20.913(2.236)^{x} $$ where \(x\) is the number of years after \(2008,\) models the sales, in millions of dollars. Use this function to estimate the sales of e-cigarettes in 2011 and in 2015 . Round to the nearest million dollars.

Price of Admission to the Magic Kingdom. In \(2015,\) the price of a one-day, one-park admission to Disney's Magic Kingdom in Florida rose to \(\$ 105 .\) The exponential function $$ D(x)=4.532(1.078)^{x} $$ where \(x\) is the number of years after \(1971,\) models the price of a ticket. (Source: AllEars.net, an independent Disney consumer website) Find the price of a ticket in \(1980,\) in \(2000,\) and in \(2012 .\) Then use the function to project the price of a ticket in 2020 .

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log (2 x+1)-\log (x-2)=1$$

Growth of a Stock. The value of a stock is given by the function $$ V(t)=58\left(1-e^{-1.1 t}\right)+20 $$ where \(V\) is the value of the stock after time \(t,\) in months. a) Graph the function. b) Find \(V(1), V(2), V(4), V(6),\) and \(V(12)\) c) After how long will the value of the stock be \(\$ 75 ?\)

Approximate the point \((s)\) of intersection of the pair of equations. $$y=\ln 3 x, y=3 x-8$$
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