To solve the equation [tex]\( e^x = 71 \)[/tex] for [tex]\( x \)[/tex], follow these steps:
1. Understand the given equation:
- We start with the exponential equation [tex]\( e^x = 71 \)[/tex], where [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
2. Isolate the variable using logarithms:
- To solve for [tex]\( x \)[/tex], we take the natural logarithm (ln), which is the inverse operation of the exponential function [tex]\( e^x \)[/tex].
3. Take the natural logarithm on both sides of the equation:
- The equation becomes [tex]\( \ln(e^x) = \ln(71) \)[/tex].
4. Apply the properties of logarithms:
- One of the properties of logarithms is that [tex]\( \ln(e^x) = x \cdot \ln(e) \)[/tex]. Since [tex]\( \ln(e) = 1 \)[/tex], the equation simplifies to [tex]\( x \cdot 1 = \ln(71) \)[/tex], or simply [tex]\( x = \ln(71) \)[/tex].
5. Evaluate the natural logarithm:
- Using a calculator or logarithm tables, we find that [tex]\( \ln(71) \approx 4.2626798770413155 \)[/tex].
Therefore, the solution to the equation [tex]\( e^x = 71 \)[/tex] is:
[tex]\[ x \approx 4.2626798770413155 \][/tex]
