Answer :
Answer:
Step-by-step explanation:
To calculate compound interest using logarithms, we use the compound interest formula:
=
(
1
+
)
A=P(1+
n
r
)
nt
Where:
A is the amount of money accumulated after
n years, including interest.
P is the principal amount (the initial amount of money, 300,000).
r is the annual interest rate (10% or 0.10).
n is the number of times that interest is compounded per year (quarterly means
=
4
n=4).
t is the time the money is invested for (5 years).
The compound interest earned
CI is given by:
=
−
CI=A−P
Using logarithms, the amount
A can be calculated as follows:
Calculate the term inside the parentheses:
1
+
=
1
+
0.10
4
=
1
+
0.025
=
1.025
1+
n
r
=1+
4
0.10
=1+0.025=1.025
Calculate the exponent:
=
4
×
5
=
20
nt=4×5=20
Use logarithms to find
A:
=
(
1.025
)
20
A=P(1.025)
20
Taking the natural logarithm (logarithm base
e) on both sides:
ln
(
)
=
ln
(
)
+
20
ln
(
1.025
)
ln(A)=ln(P)+20ln(1.025)
Now, we can plug in the values:
ln
(
)
=
ln
(
300000
)
+
20
ln
(
1.025
)
ln(A)=ln(300000)+20ln(1.025)
Let's calculate this step by step.
Find
ln
(
300000
)
ln(300000):
ln
(
300000
)
≈
12.6115
ln(300000)≈12.6115
Find
ln
(
1.025
)
ln(1.025):
ln
(
1.025
)
≈
0.0247
ln(1.025)≈0.0247
Multiply
20
20 by
ln
(
1.025
)
ln(1.025):
20
×
0.0247
=
0.494
20×0.0247=0.494
Add the results:
ln
(
)
=
12.6115
+
0.494
=
13.1055
ln(A)=12.6115+0.494=13.1055
Use the exponential function to find
A:
=
13.1055
A=e
13.1055
Let's calculate
13.1055
e
13.1055
:
≈
490
,
802.78
A≈490,802.78
Finally, the compound interest earned
CI is:
=
−
=
490
,
802.78
−
300
,
000
≈
190
,
802.78
CI=A−P=490,802.78−300,000≈190,802.78
So, the compound interest earned on
300
,
000
300,000 for 5 years at a rate of 10% per annum compounded quarterly is approximately
190
,
802.78
190,802.78.
