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Chapter 12: Problem 23

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5^{x}=17$$

Short Answer

Expert verified
The solution to the equation \(5^x = 17\) in terms of a decimal approximation correct to two decimal places is 1.76

Step by step solution

01

Conversion to Logarithmic Form

The given equation is \(5^{x}=17\). To solve for \(x\), it must be converted from exponential form to logarithmic form. Our base is 5, meaning the equivalent logarithmic equation is \(\log_5 {17} = x\).
02

Use of Calculator to Find Decimal Approximation

To find a decimal approximation of \(x\), use a calculator. Keep in mind that most calculators do not directly compute logarithms to bases other than 10 and \(e\). So we can use the change of base formula which states \(\log_b a = \frac{\log a}{\log b}\). Therefore, \(x = \frac{\log 17}{\log 5}\)
03

Calculation

Using a calculator, do the division to get an approximation of \(x\) to two decimal places. Dividing \(\log 17\) by \(\log 5\), you get approximately 1.76

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Key Concepts

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

Logarithmic Transformation
When dealing with exponential equations like the one given in the exercise, converting it into a more manageable form is crucial. This process is known as a logarithmic transformation.

Consider the equation we need to solve: \(5^{x} = 17\). Our goal is to express this equation in terms of logs. This involves recognizing the exponential expression \(5^{x}\) and converting it into its logarithmic form.
The base here is 5, and the expression is aiming to equal 17. Using the conversion formula, this translates to \(\log_5 17 = x\).

This transformation is significant because working with logarithms often simplifies the calculations involved, especially when solving for unknown exponents.
  • Helps convert complex exponentials to simpler logarithmic equations.
  • Makes solving for unknown variables, such as \(x\), more straightforward.
Change of Base Formula
Many calculators are limited to calculating logarithms in base 10 (common logarithms) or base \(e\) (natural logarithms). However, in our exercise, we need the logarithm of base 5.

This is where the change of base formula becomes very useful. This formula allows us to convert logarithms from one base to another. The formula is: \(\log_b a = \frac{\log a}{\log b}\).

Applying it to our specific case, \(\log_5 17\) can be rewritten as \(\frac{\log 17}{\log 5}\).
This way, you use the more common logarithm button found on most calculators.
  • Enables calculation of logs in uncommon bases using standard calculator functions.
  • Facilitates solving problems with exponential equations more conveniently.
Decimal Approximation
Once you've transformed the equation and adjusted the base using the change of base formula, finding a numerical solution requires calculating the decimal approximation.

For our problem, this means calculating \(\frac{\log 17}{\log 5}\). Using a calculator, you input the values for \(\log 17\) and \(\log 5\), and then perform the division.

By doing so, you get the decimal approximation of \(x\). According to the problem's instructions, you need the solution to be precise to two decimal places. After performing the calculation, you find that \(x\) is approximately 1.76.

Decimal approximation is particularly important when an exact closed form solution is challenging to interpret or apply directly. It provides a usable numerical measure of the variable.
  • Converts complex rational numbers into usable decimal format.
  • Ensures solutions are precise and practical for applications.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\)

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

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 2350 & & \text { Triple the amount invested } & 7 \\ \hline \end{array}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve. Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the \([\text { TRACE }]\) and \([\text { ZOOM }]\) features or the intersect command of your graphing utility to verify your answer.
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