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Chapter 3: Problem 58

Solve each logarithmic equation in Exercises \(49-92\). 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. $$\log _{7}(x+2)=-2$$

Short Answer

Expert verified
The solution to the logarithmic equation \(\log_{7}(x+2)=-2\) is \(x = -\frac{97}{49}\), which is approximately \(-1.98\) when rounded to two decimal places.

Step by step solution

01

Eliminate the Logarithm

To eliminate the logarithm in logarithmic equation like \(\log_{7}(x+2)=-2\), use the base of the logarithm to raise both sides. This process is known as exponentiation. This step will result in \(7^{-2} = x + 2\).
02

Solve for x

The next step is to solve for \(x\). To do this, subtract \(2\) from both sides. So, \(x = 7^{-2} - 2\).
03

Calculate the Exact Answer

After this, just calculate \(7^{-2} - 2\) to get an exact answer. This results in \(x = \frac{1}{49} - 2\). Simplifying this gives \(x=-\frac{97}{49}\).
04

Check for Domain Restrictions

To ensure that the solution fits the domain of the original logarithmic expression, one should remember that logarithms are undefined for negative numbers. The original problem has a log base 7 of \(x+2\), which means \(x+2\) must be greater than 0. Substituting -97/49 for \(x\) in \(x+2>0\) gives \(-\frac{97}{49} + 2 > 0\), which is a true statement. Hence \(x = -\frac{97}{49}\) is a valid solution.
05

Decimal Approximation

The final part of this task is to provide a decimal approximation of the solution, correct to two decimal places. So, \(-\frac{97}{49}\) approximated to two decimal places is \(-1.98\)

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Key Concepts

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

Exponentiation
Exponentiation is the process of raising a number to a power. In the context of solving logarithmic equations, it is used to eliminate the logarithm.
  • When you have an equation like \( \log_{7}(x+2) = -2 \), you can use exponentiation to "get rid" of the log by raising both sides as powers of the base of the logarithm, which in this case is 7.
  • This means you will set up an equation like \( 7^{-2} = x + 2 \).
  • This step is crucial as it transforms the logarithmic equation into a straightforward algebraic equation, making it easier to solve for the variable \(x\).
Remember, the purpose of exponentiation here is to simplify the equation and make it manageable.
Domain of Logarithmic Function
The domain of a logarithmic function is the set of all possible input values (\(x\)-values) that the function can accept. For logarithmic functions such as \( \log_b(x) \), the argument \(x\) must be greater than zero.
  • This is because you cannot take the logarithm of a negative number or zero, as these do not produce meaningful results in the context of real numbers.
  • In the problem \( \log_{7}(x+2) \), it means \(x+2 > 0\), or \(x > -2\).
Checking the solution against the domain of the original logarithmic function ensures that all solutions are valid. In this exercise, our answer \(x = -\frac{97}{49}\) satisfies this condition when substituting back into \(x+2 > 0\), confirming it's a legitimate solution.
Decimal Approximation
Decimal approximation involves converting an exact number into its decimal form, often rounded to a certain number of decimal places for simplicity.
  • Once you have found the exact solution, you may need to use a calculator to find a decimal approximation for it.
  • In this exercise, we found \(x = -\frac{97}{49}\) as the exact answer.
  • By dividing \(-97\) by \(49\), you obtain \(-1.979591836734694\).
  • For practicality and simplicity, this value is rounded to two decimal places as \(-1.98\).
This step helps in providing a more easily interpretable form of the answer, especially when dealing with real-world applications.
Exact Solution
Finding the exact solution of a mathematical problem means providing an answer that is precise and not simplified into decimal form.
  • In this exercise, solving for \(x\) using the equation \(7^{-2} = x + 2\) gives \(x = 7^{-2} - 2\).
  • This translates to \(x = \frac{1}{49} - 2\), which further simplifies to \(x = -\frac{97}{49}\).
  • An exact solution is valuable because it maintains the precision of calculations, avoiding any rounding errors that could occur if you immediately simplified to decimals.
Exact solutions are often critical in higher-level mathematics where precision is necessary.

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Most popular questions from this chapter

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\)the annual inflation rate, and \(S=\)the inflated value t years from now. Use this formula to solve. Round answers to the nearest dollar. If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 465,000\) be worth in 10 years?

Use Newton's Law of Cooling, \(T=C+\left(T_{0}-C\right) e^{k t},\) to solve this exercise. At 9: 00 A.M., a coroner arrived at the home of a person who had died. The temperature of the room was \(70^{\circ} \mathrm{F}\), and at the time of death the person had a body temperature of \(98.6^{\circ} \mathrm{F} .\) The coroner took the body's temperature at 9: 30 A.M., at which time it was \(85.6^{\circ} \mathrm{F},\) and again at 10: 00 A.M., when it was \(82.7^{\circ} \mathrm{F} .\) At what time did the person die?

The hyperbolic cosine and hyperbolic sine functions are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x=\frac{e^{x}-e^{-x}}{2} $$ a. Show that \(\cosh x\) is an even function. b. Show that \(\sinh x\) is an odd function. c. Prove that \((\cosh x)^{2}-(\sinh x)^{2}=1\)

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\)the annual inflation rate, and \(S=\)the inflated value t years from now. Use this formula to solve. Round answers to the nearest dollar. If the inflation rate is \(3 \%,\) how much will a house now worth \(\$ 510,000\) be worth in 5 years?

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