Question

how to differentiate logarithmic functions​

Answer 1

Step-by-step explanation:

First, you should know the derivatives for the basic logarithmic functions:

\dfrac{d}{dx}\ln(x)=\dfrac{1}{x}

dx

d

ln(x)=

x

1

start fraction, d, divided by, d, x, end fraction, natural log, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, x, end fraction

\dfrac{d}{dx}\log_b(x)=\dfrac{1}{\ln(b)\cdot x}

dx

d

log

b

(x)=

ln(b)⋅x

1

start fraction, d, divided by, d, x, end fraction, log, start base, b, end base, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, natural log, left parenthesis, b, right parenthesis, dot, x, end fraction

Notice that \ln(x)=\log_e(x)ln(x)=log

e

(x)natural log, left parenthesis, x, right parenthesis, equals, log, start base, e, end base, left parenthesis, x, right parenthesis is a specific case of the general form \log_b(x)log

b

(x)log, start base, b, end base, left parenthesis, x, right parenthesis where b=eb=eb, equals, e. Since \ln(e)=1ln(e)=1natural log, left parenthesis, e, right parenthesis, equals, 1 we obtain the same result.

You can actually use the derivative of \ln(x)ln(x)natural log, left parenthesis, x, right parenthesis (along with the constant multiple rule) to obtain the general derivative of \log_b(x)log

b

(x)log, start base, b, end base, left parenthesis, x, right parenthesis.