Problem 23
Question
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. $$5^{x}=17$$
Step-by-Step Solution
Verified Answer
The solution to the equation \(5^x = 17\) in terms of a decimal approximation correct to two decimal places is 1.76
1Step 1: 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 \(\log_5 {17} = x\).
2Step 2: 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, \(x = \frac{\log 17}{\log 5}\)
3Step 3: 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
Key Concepts
Logarithmic TransformationChange of Base FormulaDecimal Approximation
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.
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.
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.
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.
Other exercises in this chapter
Problem 22
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calcula
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Graph each function by making a table of coordinates. If applicable, use a graphing unility to confirm your hand-drawn graph. $$h(x)=\left(\frac{1}{3}\right)^{x
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Evaluate each expression without using a calculator. $$\log _{2} 64$$
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Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calcula
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