Chapter 18: Problem 16
Solve for \(x\) to three significant digits. $$5^{2 x}=3^{3 x+1}$$
Short Answer
Expert verified
After calculating, the approximate value of x is \( x \approx -1.37 \).
Step by step solution
01
Express the Equation with Logarithms
To solve for the variable when it is an exponent in both sides of the equation, we take the logarithm of both sides, which yields \(\log(5^{2x}) = \log(3^{3x+1})\).
02
Apply Logarithmic Properties
Use the property of logarithms that states \(\log(a^b) = b \cdot \log(a)\) to bring the exponents down: \(2x \cdot \log(5) = (3x + 1) \cdot \log(3)\).
03
Distribute the Logarithms
Distribute the logarithms across the terms on the right side, yielding \(2x \cdot \log(5) = 3x \cdot \log(3) + \log(3)\).
04
Isolate the Variable
Bring all terms containing \(x\) to one side and constants to the other, resulting in \(2x \cdot \log(5) - 3x \cdot \log(3) = \log(3)\).
05
Factor Out the Variable
Factor \(x\) out from the left-hand side: \(x(2 \cdot \log(5) - 3 \cdot \log(3)) = \log(3)\).
06
Solve for the Variable
Divide both sides by \(2 \cdot \log(5) - 3 \cdot \log(3)\) to solve for \(x\): \(x = \frac{\log(3)}{2 \cdot \log(5) - 3 \cdot \log(3)}\).
07
Calculate the Numerical Value
Use a calculator to find the numerical value of \(x\) to three significant digits. After evaluating the right-hand side, you get the approximate value of \(x\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Properties
Understanding logarithmic properties is crucial when solving exponential equations like the one in the exercise provided, \(5^{2x} = 3^{3x+1}\). The fundamental property used in solving such an equation is that the logarithm of a powered number, such as \(a^b\), can be rewritten as the exponent times the logarithm of the base: \(\log(a^b) = b \cdot \log(a)\). This property allows us to bring down exponents and make the variables more accessible.
Additionally, knowing that logarithms convert the operation of multiplication into addition and division into subtraction can significantly simplify the process of solving equations. For example, if we have \(\log(ab)\), this can be separated into \(\log(a) + \log(b)\), and likewise, \(\log(\frac{a}{b})\) can be rewritten as \(\log(a) - \log(b)\). These properties are applied when distributing logarithms across terms or when combining terms into a single logarithm.
Furthermore, the bases of the logarithms are often assumed to be 10 in common logarithms or \(e\) in natural logarithms unless specified otherwise. In the step-by-step solution, using logarithms with base 10 simplifies the equation, making the unknown variable \(x\) solvable.
Additionally, knowing that logarithms convert the operation of multiplication into addition and division into subtraction can significantly simplify the process of solving equations. For example, if we have \(\log(ab)\), this can be separated into \(\log(a) + \log(b)\), and likewise, \(\log(\frac{a}{b})\) can be rewritten as \(\log(a) - \log(b)\). These properties are applied when distributing logarithms across terms or when combining terms into a single logarithm.
Furthermore, the bases of the logarithms are often assumed to be 10 in common logarithms or \(e\) in natural logarithms unless specified otherwise. In the step-by-step solution, using logarithms with base 10 simplifies the equation, making the unknown variable \(x\) solvable.
Isolate the Variable
Isolating the variable is an essential technique used in algebra to solve equations. It involves manipulating the equation to get the variable we're solving for by itself on one side of the equality. In our exercise's context, with \(5^{2x} = 3^{3x+1}\), after taking the logarithm of both sides and applying logarithmic properties, we reach a stage where the variable \(x\) is within multiple terms.
To isolate \(x\), we need to collect all terms involving the variable on one side of the equation and move constant terms to the other side. This is evident in the step where we subtract \(3x \cdot \log(3)\) from both sides to yield \(2x \cdot \log(5) - 3x \cdot \log(3) = \log(3)\). Once there, we can factor out \(x\) from the left-hand side, which demonstrates another strategy: factoring common variables from multiple terms to further isolate the variable.
Finally, we divide both sides of the equation by the remaining coefficient of \(x\), which, in this case, is \(2 \cdot \log(5) - 3 \cdot \log(3)\). This step effectively leaves \(x\) isolated, allowing us to solve for its numerical value using the logarithm values calculated or retrieved from a calculator.
To isolate \(x\), we need to collect all terms involving the variable on one side of the equation and move constant terms to the other side. This is evident in the step where we subtract \(3x \cdot \log(3)\) from both sides to yield \(2x \cdot \log(5) - 3x \cdot \log(3) = \log(3)\). Once there, we can factor out \(x\) from the left-hand side, which demonstrates another strategy: factoring common variables from multiple terms to further isolate the variable.
Finally, we divide both sides of the equation by the remaining coefficient of \(x\), which, in this case, is \(2 \cdot \log(5) - 3 \cdot \log(3)\). This step effectively leaves \(x\) isolated, allowing us to solve for its numerical value using the logarithm values calculated or retrieved from a calculator.
Significant Digits
Significant digits (also known as significant figures) are a key concept when reporting the results of calculations in science and engineering because they represent the precision of a measurement. The number of significant digits in a numerical result indicates the certainty of the measurement, with more significant digits implying higher precision.
In our exercise, the -problem requires the solution to be rounded to three significant digits. This means that, after calculating the value of \(x\) using a calculator, the result should be reported with the most significant three figures. For instance, if the calculator shows an answer such as 0.12345, the number 0.123 is rounded according to the fourth digit after the decimal point – since the fourth digit is a '4,' the value of \(x\) should be reported as 0.123.
Being conscientious about significant digits is not just about truncating or rounding numbers; it's also about conveying the accuracy of the computed results. When using a calculator, it often displays more digits than are significant for the problem, so it's essential to round to the appropriate number of significant digits to avoid overrepresenting the precision of the solution.
In our exercise, the -problem requires the solution to be rounded to three significant digits. This means that, after calculating the value of \(x\) using a calculator, the result should be reported with the most significant three figures. For instance, if the calculator shows an answer such as 0.12345, the number 0.123 is rounded according to the fourth digit after the decimal point – since the fourth digit is a '4,' the value of \(x\) should be reported as 0.123.
Being conscientious about significant digits is not just about truncating or rounding numbers; it's also about conveying the accuracy of the computed results. When using a calculator, it often displays more digits than are significant for the problem, so it's essential to round to the appropriate number of significant digits to avoid overrepresenting the precision of the solution.
