Working rule to find the derivative of a general logarithmic function :
Find ᵈʸ⁄dₓ, 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.
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(sin√x)
Solution :
y = log5(sin√x)
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