#color(blue)("Introduction of concepts"#
Example of principle: Try this on your calculator
Using log to base 10 enter #log(10)# and you get the answer of 1.
Log to base e is called 'natural' logs and is written as #ln(x)# for any value #x#
#color(brown)("Consequently "ln(e)=1)# Try that on your calculator
[ you may have to enter #ln(e^1)# ]
Another trick is that #log(x^2) -> 2log(x) => ln(x^2)=2ln(x)#
Combining these two ideas:
#ln(e^2)" "=" "2ln(e)" "=" "2xx1=2#
#color(brown)("So "ln(e^x)" "=" "xln(e)" "=" "x xx1" " =" " x)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Solving the question")#
Given:#" "e^x+3=6#
Subtract 3 from both sides
#" "e^x=6-3#
#" "e^x=3#
Take logs of both sides
#" "ln(e^x)=ln(3)#
#" "xln(e)=ln(3)#
But #ln(e)=1# giving
#color(green)(x=ln(3) ~~1.099" to 3 decimal places")#
