Problem 34
Question
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. $$e^{1-8 x}=7957$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x \approx -1.92\)
1Step 1: Write down the given equation
The equation provided is \(e^{1-8x}=7957\).
2Step 2: Use natural logarithm to both sides of equation
Taking the natural logarithm to both sides of the equation we get \(\ln(e^{1-8x}) = \ln(7957)\). The left side simplifies to \(1-8x\) due to the property of the natural logarithm.
3Step 3: Solve for 'x'
To isolate variable 'x', we first add \(8x\) to both sides to get \(1=\ln(7957)+8x\). Then subtract \(\ln(7957)\) from both sides to get \(8x=1-\ln(7957)\). Finally, divide both sides by 8 to solve for 'x', yielding to \(x = \frac{1-\ln(7957)}{8}\).
4Step 4: Find the decimal approximation
Use a calculator to find the decimal value of 'x', yielding to \(x≈-1.92\) to two decimal places.
Other exercises in this chapter
Problem 34
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