Problem 36
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. $$e^{1-8 x}=7957$$
Step-by-Step Solution
Verified Answer
The solution is \(x = - (ln(7957) - 1)/8\), which can be approximated to a decimal using a calculator.
1Step 1: Express the given equation in exponential form
The given equation is \(e^{1-8x}=7957\). Here, \(e\) is the base of the natural logarithm, \(1-8x\) is the exponent and \(7957\) is the value of the exponential function.
2Step 2: Take logarithm on both sides
Taking the natural logarithm (logarithm to the base \(e\)) on both sides of the equation. We have: \(ln(e^{1-8x})=ln(7957)\). Using the property of logarithms that the logarithm of an exponent is the product of the exponent and the logarithm of the base, we simplify the left-hand side: \((1-8x)ln(e)=ln(7957)\)
3Step 3: Simplify the equation
Since the natural logarithm of \(e\) is 1 (\(ln(e)=1\)), the equation simplifies to: \(1-8x=ln(7957)\)
4Step 4: Isolate the variable \(x\)
Isolate the variable \(x\) by subtracting 1 from both sides and then dividing by -8: \(x = - (ln(7957) - 1)/8\)
5Step 5: Use a calculator to obtain a decimal approximation
Put this expression in a calculator to obtain the decimal approximation to two decimal places.
Key Concepts
Natural LogarithmSolving EquationsDecimal ApproximationExponential Function
Natural Logarithm
The natural logarithm, often denoted as \(ln\), is a special type of logarithm where the base is Euler's number, \(e\). This constant, approximately 2.71828, is fundamental in mathematics, especially for growth processes. When you see \(ln(x)\), it asks: "To what power must \(e\) be raised to get \(x\)?"
In our problem, we used the natural logarithm to facilitate handling the exponent in the equation \(e^{1-8x} = 7957\). By applying the natural logarithm to both sides, \(ln(e^{1-8x}) = ln(7957)\), we took advantage of the properties of logarithms. The key property says that \(ln(a^b) = b \cdot ln(a)\), allowing us to move the exponent out in front, simplifying the expression.
In our problem, we used the natural logarithm to facilitate handling the exponent in the equation \(e^{1-8x} = 7957\). By applying the natural logarithm to both sides, \(ln(e^{1-8x}) = ln(7957)\), we took advantage of the properties of logarithms. The key property says that \(ln(a^b) = b \cdot ln(a)\), allowing us to move the exponent out in front, simplifying the expression.
Solving Equations
Solving equations involves finding the unknown that makes the equation true. In exponential equations, especially those with \(e\) as the base, the natural logarithm is a powerful tool. By taking the \(ln\) of both sides, we can transform the original equation into a simpler, linear form.
For \(e^{1-8x} = 7957\), once the natural logarithm is applied, it becomes easier to isolate \(x\), the variable of interest. This requires strategic algebraic manipulations that typically include:
For \(e^{1-8x} = 7957\), once the natural logarithm is applied, it becomes easier to isolate \(x\), the variable of interest. This requires strategic algebraic manipulations that typically include:
- Applying \(ln\) to both sides to remove the variable from the exponent.
- Using properties of logarithms to simplify expressions.
- Rearranging terms to isolate the variable.
Decimal Approximation
Decimal approximation is crucial when you need a numerical answer rather than an exact expression involving logarithms. Many real-world problems are solved using approximate values, which are easier to interpret and apply.
After isolating \(x\) using \(- (ln(7957) - 1)/8\), a calculator is utilized to find this expression's decimal form. The answer, when rounded to two decimal places, provides a practical solution that is often necessary for accurate measurement in experiments or predictions.
This method highlights how mathematical concepts bridge abstract calculations and concrete applications.
After isolating \(x\) using \(- (ln(7957) - 1)/8\), a calculator is utilized to find this expression's decimal form. The answer, when rounded to two decimal places, provides a practical solution that is often necessary for accurate measurement in experiments or predictions.
This method highlights how mathematical concepts bridge abstract calculations and concrete applications.
Exponential Function
Exponential functions are a central concept in mathematics, defined as functions where a constant base is raised to a variable exponent. They exhibit rapid growth or decay, modeling phenomena such as population dynamics and radioactive decay.
In the equation \(e^{1-8x} = 7957\), \(e^{1-8x}\) represents an exponential function with \(e\) as the base. The significance of the base \(e\) lies in its inherent natural growth property, which makes exponential functions with this base predominant in many scientific fields.
Understanding exponential functions requires recognizing behaviors such as:
In the equation \(e^{1-8x} = 7957\), \(e^{1-8x}\) represents an exponential function with \(e\) as the base. The significance of the base \(e\) lies in its inherent natural growth property, which makes exponential functions with this base predominant in many scientific fields.
Understanding exponential functions requires recognizing behaviors such as:
- Multiplicative growth or decay over equal intervals.
- Sensitivity to changes in the exponent.
- Asymptotic behavior, meaning increasingly approaching a boundary but never reaching it.
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