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Chapter 4: Problem 79

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (3 x-3)=\log (x+1)+\log 4$$

Short Answer

Expert verified
The solution \(x=-7\) doesn't satisfy the original logarithmic equation and is not in the domain of the original logarithmic expressions. Therefore, the given logarithmic equation has no solutions.

Step by step solution

01

Combine the right side of the equation

Since we know that the sum of two logarithms equals the logarithm of the product of their arguments, we can apply this property to combine the two logarithms on the right-hand side of our equation. Using the logarithmic property \( \log a + \log b = \log (ab) \) we get: \n\n\( \log (3x - 3) = \log [(x+1) * 4] \)
02

Simplify the equation

Next, we simplify the right side of the equation by multiplying the arguments together to get: \n\n\( \log (3x - 3) = \log (4x + 4) \)
03

Solve for \( x \)

Since the base of the logarithm on both sides of the equation is the same, we can equate their arguments to make the equation simpler. This gives us the equation: \n\n\( 3x - 3 = 4x + 4 \). Solution of this linear equation results in \( x = -7 \)
04

Validate the solution

We must verify if \( x = -7 \) satisfies the original equation and does not lead to taking a logarithm of a negative number. Logarithms of negative numbers are undefined. Substituting \( x = -7 \) into the original equation yields \( \log(-21) = \log(-3) + \log 4 \), so the solution is not valid because it does not satisfy the original equation.

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Most popular questions from this chapter

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. $$\log _{3}(4 x-7)=2$$

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

Explain why the logarithm of 1 with base \(b\) is \(0 .\)

Research applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, human memory, progress over time in a sport, profit over time. For the area that you select, explain how logarithmic functions are used and provide examples.

What question can be asked to help evaluate \(\log _{3} 81 ?\)
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