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Chapter 4: Problem 99

Solve each equation. $$ \ln (2 x+1)+\ln (x-3)-2 \ln x=0 $$

Short Answer

Expert verified
The solution to the equation is \(x=1\).

Step by step solution

01

Use properties of logarithms

Using the property \(\ln(a)+\ln(b)=\ln(ab)\) and \(\ln(c)-\ln(d)=\ln(\frac{c}{d})\), the given equation can be rewritten as \(\ln((2x+1)(x-3)/x^2)=0\)
02

Express the logarithm as an exponent

For any \(b^y=x\), it can be rewritten as \(\ln_b(x)=y\). Therefore, you can rewrite the equation from step 1 as \(((2x+1)(x-3)/x^2)=e^0\)
03

Simplify the equation

As \(e^0=1\), the equation from step 2 simplifies to \(((2x+1)(x-3))/x^2=1\). Multiply both sides by \(x^2\) to get \((2x+1)(x-3)=x^2\)
04

Continue simplifying

Expand the left-hand side to get \(2x^2+2x-3x-3=x^2\). Simplify further to \(x^2+2x-3=0\)
05

Solve for x

You can solve for x by factoring, completing the square or using the quadratic formula. After factoring, the equation becomes \((x+3)(x-1)=0\). Setting each factor equal to zero gives two potential solutions: \(x=-3\) or \(x=1\)
06

Check the solutions

Substitute the solutions into the original equation to check if they are valid solutions. \(x=-3\) makes the argument of the second logarithm negative, which is not possible. So, \(x=-3\) is not a solution. However, \(x=1\) is a valid solution.

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Most popular questions from this chapter

If \(f(x)=\log _{b} x,\) show that $$ \frac{f(x+h)-f(x)}{h}=\log _{b}\left(1+\frac{h}{x}\right)^{\frac{1}{h}} h \neq 0 $$

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$ y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2} $$

Given \(f(x)=\frac{2}{x+1}\) and \(g(x)=\frac{1}{x},\) find each of the following: a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) (Section \(2.6, \text { Example } 6)\)

Consider the quadratic function $$f(x)=-4 x^{2}-16 x+3$$ a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. (Section 3.1, { Example } 4)

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.
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