At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
