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Chapter 3: Problem 37

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$7^{x+2}=410$$

Short Answer

Expert verified
The value of \(x\) is, approximately, the result from the calculated operation \(\frac{ln(410)}{ln(7)} - 2\). This result must be approximated to two decimal places by utilizing a calculator.

Step by step solution

01

Isolate the exponential term

The given equation is \(7^{x+2}=410\). To isolate the exponential term, there's no need for any adjustments because \(7^{x+2}\) is already on one side of the equation.
02

Apply the logarithm

To solve for the exponent \(x+2\), take the natural logarithm, or common logarithm on both sides. For example, applying natural log (ln) we get, \(ln(7^{x+2}) = ln(410)\). Using the power rule of logarithms, the right-hand side remains the same, but the left-hand side can be written as \((x+2)*ln(7)\). Thus, the equation becomes, \((x+2)*ln(7) = ln(410)\).
03

Solve for x

Finally, manipulate the equation to solve for \(x\). First, divide both sides by \(ln(7)\) to get \(x+2 = \frac{ln(410)}{ln(7)}\). Then, subtract 2 from both sides to get \(x = \frac{ln(410)}{ln(7)} - 2\).
04

Approximate using a calculator

Now, use a calculator to carry out the divisions and subtraction in the last equation and estimate \(x\), rounded to two decimal places. This will be our numerical solution.

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Key Concepts

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

Logarithms
Logarithms are essential tools in mathematics and are often used to solve exponential equations, like the example from the exercise. Simply put, the logarithm helps to find the exponent that raises a number (the base) to a specific value. For instance, in the equation \(7^{x+2} = 410\), the exponent \(x+2\) is what we're solving for.

Logarithms have a few key properties that make them powerful:
  • Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
  • Quotient Rule: \( \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \)
  • Power Rule: \( \log_b(M^k) = k \cdot \log_b(M) \)
By using these properties, logarithms can simplify the process of solving exponential equations. For our exercise, we apply the power rule, allowing us to bring the exponent down and solve for \(x\). This conversion is a critical step in dealing with exponential equations.
Natural Logarithm
The natural logarithm is a specific kind of logarithm, usually denoted as \( \ln \). It uses the constant \( e \) (approximately 2.71828) as its base. Natural logarithms are widely used in scientific calculations due to the unique properties of \( e \), especially when dealing with exponential growth and decay.

In the exercise, when solving \(7^{x+2} = 410\), we can choose to apply the natural logarithm. Using the logarithm base \( e \) simplifies calculations involving ratios, a common occurrence in natural processes:
  • For example, taking natural logs of both sides gives: \( \ln(7^{x+2}) = \ln(410) \).
  • Using the power rule, this further simplifies to \( (x+2)\ln(7) = \ln(410) \).
Choosing \( \ln \) is often a matter of preference or context, depending heavily on what specific calculations need to be simplified or analyzed.
Common Logarithm
The common logarithm is another frequently used type, denoted as \( \log \) or \( \log_{10} \), with 10 as its base. This logarithm is especially useful in situations involving orders of magnitude, such as pH calculations, decibels, or any tenfold increase in a value.

In our exercise, instead of using the natural logarithm, we could choose the common logarithm:
  • Taking the common logarithm of both sides of \(7^{x+2} = 410\) results in \( \log(7^{x+2}) = \log(410) \).
  • Using the power rule, this equation becomes \( (x+2)\log(7) = \log(410) \).
Calculating with \( \log \) will yield the same final answer as using \( \ln \), although the actual number for each logarithm will differ slightly due to the different bases. Opt for \( \log \) if you鈥檙e dealing with everyday exponential differences.

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Most popular questions from this chapter

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

The \(p H\) scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a pH greater than 7. The lower the \(p H\) below 7 , the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. (GRAPH CAN'T COPY). The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. Normal, unpolluted rain has a pH of about 5.6. What is the hydrogen ion concentration? b. An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? c. How many times greater is the hydrogen ion concentration of the acidic rainfall in part (b) than the normal rainfall in part (a)?

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

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. $$2^{x+1}=8$$

graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f. $$f(x)=\ln x, g(x)=\ln x+3$$
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