Solved Math Problems on Logarithm and Exponential Functions

This document provides step-by-step solutions to math problems on logarithm and exponential functions. It includes problems on population growth, logarithmic equations, and more.

1)
Answer :
Population of West Goma (in millions):
f (x) = 16.7e0.0015x
1

Population of East Goma (in millions):
g(x) = 14.6e0.0135x
The point where both the population will be equal, we have:
16.7e0.0015x= 14.6e0.0135x
16.7
14.6= e0.0135x
e0.0015x
[Property of exponents :
ea
eb = ea−b]
∴ 16.7
14.6= e0.012x
[Taking log on both sides]
ln 16.7
14.6 = 0.012x
=⇒ x = 0.1344
0.012= 11.1989
The population ofWest and East Goma willbe equalat the point x =
11.1989.Since, x = 0 corresponds to year 1998, x = 11.1989 will correspond
to year 1998 + 11.1989 = 2009.1989.
Thus the population of East and West Goma will be equal in year 2010.
2)
Answer
t = log6 36
[Take both sides as exponents to base 6]
6t = 6log6 36
[Property of exponential : aloga b = b]
∴ 6 t = 36
=⇒ 6t − 36 = 0
3)
Answer:
2

log6
3
2
[Property of logarithm : loga (c/d) = loga c − loga d]
=⇒ log6 3 − log6 2
4)
Answer :
log (x + 9) = 1 − log x
[Add log x to both sides]
log (x + 9) + log x = 1
[Property of logarithm : log (c × d) = log c + log d]
log (x(x + 9)) = 1
[Express : 1 = log 10]
log (x2 + 9x) = log 10
[Take both sides out of the log]
x2 + 9x = 10
=⇒ x 2 + 9x − 10 = 0
x2 + 10x − 1x − 10 = 0
x(x + 10) − 1(x + 10) = 0
(x + 10)(x − 1) = 0
=⇒ x = −10 or 1
The root x = −10 is not acceptable since log ofnegative numbers in not
defined.Therefore, the solution is:
x = 1
3

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