# DERIVATIVE OF A GENERAL LOGARITHMIC FUNCTION

Working rule to find the derivative of a general logarithmic function :

Find ᵈʸ⁄d, if y = logau, where u is a function of x.

Step 1 :

y = logau

Convert the above equation to exponential form.

ay = u

Step 2 :

Take natural logarithm on both sides.

ln(ay) = ln(u)

Step 3 :

Use the power rule of logarithm.

yln(a) = ln(u)

Step 4 :

Find the derivative on both sides.

## Solved Problems

Find ᵈʸ⁄dₓ in each of the following.

Problem 1 :

y = log6x

Solution :

y = log6x

Problem 2 :

y = log10(2x)

Solution :

﻿Method 1 :﻿

y = log10(2x)

﻿Method 2 :﻿

y = log10(2x)

Use the product rule of logarithm.

y = log102 + log10x

Problem 3 :

y = log5(3x2)

Solution :

﻿Method 1 :﻿

y = log5(3x2)

﻿Method 2 :﻿

y = log5(3x2)

Use the product rule of logarithm.

y = log53 + log5x2

Use the power rule of logarithm.

y = log53 + 2log5x

Problem 4 :

y = log10√x

Solution :

Method 1 :

y = log10√x

Method 2 :

y = log10√x

y = log10x½

Use the power rule of logarithm.

Problem 5 :

y = log10ex

Solution :

Method 1 :

y = log10ex

Method 2 :

y = log10ex

Use the power rule of logarithm.

y = xlog10e

Problem 6 :

y = log2(3x2 + 5x - 8)

Solution :

y = log2(3x2 + 5x - 8)

Problem 7 :

y = log7(sinx)

Solution :

y = log7(sinx)

Problem 8 :

y = log4(cosx)

Solution :

y = log4(cosx)

Problem 9 :

y = log9(tanx)

Solution :

y = log9(tanx)

Problem 10 :

y = log5(sinx)

Solution :

y = log5(sinx)

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