To solve the equation [tex]\( e^x = 70 \)[/tex] for [tex]\( x \)[/tex], follow these steps:
1. Understand the equation: The equation given is an exponential equation where the base of the exponent is the natural number [tex]\( e \)[/tex] (i.e., Euler's number, approximately equal to 2.71828).
2. Isolate the exponential expression: The equation is already in the form where the exponential expression [tex]\( e^x \)[/tex] is isolated on one side of the equation.
3. Apply the natural logarithm: To solve for [tex]\( x \)[/tex], take the natural logarithm (denoted as [tex]\(\ln\)[/tex]) of both sides of the equation. The natural logarithm is the inverse function of the exponential function.
[tex]\[
\ln(e^x) = \ln(70)
\][/tex]
4. Simplify using properties of logarithms: Use the property of logarithms that states [tex]\(\ln(e^x) = x \cdot \ln(e)\)[/tex]. Since [tex]\(\ln(e) = 1\)[/tex], this simplifies to:
[tex]\[
x \cdot \ln(e) = \ln(70)
\][/tex]
[tex]\[
x \cdot 1 = \ln(70)
\][/tex]
[tex]\[
x = \ln(70)
\][/tex]
Therefore, the solution to the equation [tex]\( e^x = 70 \)[/tex] is:
[tex]\[
x = \ln(70)
\][/tex]
