How to solve this problem by using logarithm:
David inherits $10000 and invests in a Guaranteed Investment Certificate (GIC) that earns 6%, compounded semi-annually. How long will it take for the GIC to be worth $11000.
10000(1.03)^n = 11000 , where n is half=years
1.03^n = 1.1
n log 1.03 = log 1.1
n = log 1.1/log 1.03 = 3.224 half years
it will take 1.612 years or about 1 1/2 years
To solve this problem using logarithm, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
In this case, David invests $10000, the interest rate is 6%, compounded semi-annually, and he wants the GIC to be worth $11000.
Let's substitute the values into the formula:
11000 = 10000(1 + 0.06/2)^(2t)
Now, let's solve for t by using logarithms. Since the exponential term is the only thing with the variable, we can isolate it:
(1 + 0.06/2)^(2t) = 11000/10000
(1 + 0.03)^(2t) = 1.1
Taking the logarithm (base 10) of both sides of the equation gives:
log((1 + 0.03)^(2t)) = log(1.1)
Using the property of logarithms that states log(a^b) = b * log(a), we can simplify the equation:
2t * log(1 + 0.03) = log(1.1)
Now, divide both sides of the equation by 2 * log(1.03):
t = log(1.1) / (2 * log(1 + 0.03))
Using a calculator or computer, evaluate the right side of the equation to find the value of t.
To solve this problem using logarithms, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the initial principal amount (the inheritance)
r = the annual interest rate (expressed as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, we know that David's initial investment (P) is $10000, the interest rate (r) is 6% (or 0.06), the interest is compounded semi-annually (n = 2), and we want to find the time (t) it takes for the investment to be worth $11000.
Substituting these values into the formula, we get:
11000 = 10000(1 + 0.06/2)^(2t)
To solve for t, we will need to use logarithms. Taking the natural logarithm (ln) of both sides of the equation:
ln(11000) = ln(10000(1 + 0.06/2)^(2t))
Using the properties of logarithms, we can simplify the equation:
ln(11000) = ln(10000) + ln(1 + 0.06/2)^(2t)
ln(11000) = ln(10000) + 2t * ln(1 + 0.06/2)
Now, isolate t by subtracting ln(10000) from both sides:
ln(11000) - ln(10000) = 2t * ln(1 + 0.06/2)
Using the logarithmic identity ln(a) - ln(b) = ln(a/b), we can further simplify:
ln(11000/10000) = 2t * ln(1 + 0.06/2)
Simplifying the left side:
ln(1.1) = 2t * ln(1.03)
Now, divide both sides of the equation by 2 ln(1.03) to solve for t:
t = ln(1.1) / (2 * ln(1.03))
Using a calculator, we can evaluate this expression:
t ≈ 6.433
Therefore, it will take approximately 6.433 years for the GIC to be worth $11000.