To solve the exponential equation [tex]\( e^x = 17 \)[/tex], we can use natural logarithms to simplify it.
Step 1: Recall the equation we need to solve:
[tex]\[ e^x = 17 \][/tex]
Step 2: Take the natural logarithm (denoted as [tex]\( \ln \)[/tex]) of both sides of the equation. The natural logarithm helps to simplify the equation because it is the inverse function of the exponential function.
Taking the natural logarithm of both sides:
[tex]\[ \ln(e^x) = \ln(17) \][/tex]
Step 3: Use the property of natural logarithms that states [tex]\( \ln(e^x) = x \)[/tex]. This property works because the natural log base [tex]\( e \)[/tex] of [tex]\( e \)[/tex] raised to any power is simply that power.
So, applying this property:
[tex]\[ x = \ln(17) \][/tex]
Step 4: Now, we have [tex]\( x \)[/tex] in terms of the natural logarithm of 17. Using a calculator or appropriate mathematical tool, we find that:
[tex]\[ x = \ln(17) \approx 2.833213344056216 \][/tex]
Therefore, the solution to the equation [tex]\( e^x = 17 \)[/tex] is:
[tex]\[ x \approx 2.833213344056216 \][/tex]
