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Chapter 12: Problem 121

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\).

Short Answer

Expert verified
The solution for the first equation \(\log _{3}(x-1)=4\) is \(x=3^4 + 1 = 82\). The solution for the second equation \(\log _{3}(x-1)=\log _{3} 4\) is \(x = 4 + 1 = 5\). The first equation required conversion to an exponential equation for solving, while the second equation could be solved by directly equating the quantities within the logarithms, due to the property of logarithms.

Step by step solution

01

Solving the First Equation

At the core, the equation \(\log _{3}(x-1)=4\) tells us that the base 3 raised to the power 4 gives us the quantity \((x-1)\). Hence, our first step is to convert this logarithmic expression into an exponential one, which would be \(3^4=x-1\). Solving this gives us \(x=3^4 + 1\).
02

Solving the Second Equation

On the other hand, the equation \(\log _{3}(x-1)=\log _{3} 4\) implies that the quantity \((x-1)\) and the number 4 are equivalent. This is due to the property of logarithms that if \(\log _{b} a = \log _{b} c\), then \(a = c\). Hence, here we can immediately equate \(x-1\) to 4, and solve to find that \(x = 4 + 1\).
03

Contrasting the Two Methods

As can be observed, the primary difference between solving these two equations lies in the direct application of the property of logarithms in the second equation, as opposed to the conversion from logarithmic to exponential form in the first. Understanding the appropriate situations to apply these different approaches is fundamental in solving logarithm-based problems effectively.

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Most popular questions from this chapter

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

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