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Chapter 4: Problem 34

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. $$e^{1-8 x}=7957$$

Short Answer

Expert verified
The solution to the equation is \(x \approx -1.92\)

Step by step solution

01

Write down the given equation

The equation provided is \(e^{1-8x}=7957\).
02

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.
03

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 \(x = \frac{1-\ln(7957)}{8}\).
04

Find the decimal approximation

Use a calculator to find the decimal value of 'x', yielding to \(x鈮�-1.92\) to two decimal places.

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