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# Mathematics

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4. Solve:
a Solve for x if 2logx = 2 + log25
b Simplify log3 81 - log3 27 + 4log3 v3
c Find the inverse of the function f(x) = e^x + 1

5. This is completing a piecewise function which consists of 2 straight lines and a parabola.
Coordinates given are (-6,0) (-3,6)-straight line, (-3,6) (.5, -6) (2,-4) -parabola, and last straight line (2,-4) parallel to x-axis
Need equation and domain for each piece

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Solution1a:
Given: 2logx = 2 +log25
log100 = 2 [Since, log100 = log102 = 2log10 =2]
Therefore,
2logx = log100 +log25
Using power law of logarithms, 2log(x) = log(x2)
Therefore,
log(x2) = log(25) + log(100)

Using sum rule for logarithms, log(25) + log(100) = log(25x100):
log(x2) = log(2500)

Removing all logarithms (by raising everything to the power of 10):
x2= 2500
Or, x = +/- 50

Since 2log(-50) doesn't exist, therefore x = +50

Solution1b:
log3 81 = log81/log3 [Using the change of base rule for logarithms]
= log 34/log3
= 4log3/log3 [Using power law of ...

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• MSc, Osmania University
• MSc, Indian Institute of Technology - Roorkee (I.I.T.-ROORKEE)
• BSc, Banaras Hindu University
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