Verify properties of natural log. One way to define the natural log is ln x = in
ID: 2883826 • Letter: V
Question
Verify properties of natural log. One way to define the natural log is ln x = integral^x_1 dt/t This definition is sufficient for most properties that we are used to: (1) ln(1) = 0 (2) ln (x/y) = ln x - ln y (3) ln (1/x) = -ln x (4) ln (xy) = ln x + ln y (5) ln(x^r) = r ln x Begin by verifying property 1 using properties of definite integrals. The key to verifying properties 2, 4, 5 is to start with the quantity on the left and express it as an integral. Followed by making the appropriate substitution to handle the bounds. Property 3 follows from properties 1 and 2. Warning: In order to receive credit you must verify the properties using the integral definition.Explanation / Answer
given
1/t dt for t = ( 1 to x )
we know Use the common integral 1/t dt = ln ( t ) + C
apply limits
ln ( t ) for t = ( 1 to x )
ln ( x ) - ln ( 1 )
we know ln ( 1 ) = 0
therefore answer is ln(x)
L.H.S = R.H.S
ln(x) = 1/t dt for t = ( 1 to x )
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