use logarithm to solve
A car depreciated at 15% per year. How long is it until it is worth half the original value?
each year the value is 0.85 as much. So, after t years it is worth .85^t as much. So, you want t when
.85^t = 0.5
Now take the log of both sides:
t log .85 = log .5
t = (log .5)/(log .85) = 4.26
An automobile depreciates by 23% per year. How soon will it be worth only half its original value?
To solve this problem using logarithms, we can use the formula for exponential decay:
A = P(1 - r)^t
Where:
A = Final value (half the original value)
P = Initial value (original value)
r = Rate of decay (15% or 0.15)
t = Time (in years)
We want to find the value of t when A is half the initial value, so we can rewrite the equation as:
1/2 = (1 - 0.15)^t
To isolate the exponent, we can take the logarithm of both sides. Let's use the natural logarithm (ln):
ln(1/2) = ln((1 - 0.15)^t)
Using the property of logarithms, we can bring down the exponent:
ln(1/2) = t * ln(1 - 0.15)
Now, we can solve for t by dividing both sides by ln(1 - 0.15):
t = ln(1/2) / ln(1 - 0.15)
Calculating this on a calculator, we get:
t ≈ 4.95 years
Therefore, it will take approximately 4.95 years for the car to be worth half its original value.
To solve this problem using logarithm, we can use the formula for exponential decay:
A = P(1 - r)^t
Where:
A = Final value
P = Initial value
r = Rate of decrease (as a decimal)
t = Time (in years)
In this case, the car is worth half the original value, so A = 0.5P. The rate of decrease is 15% per year, or 0.15, and we need to find the time it takes until the car is worth half its original value.
Substituting the given values into the formula, we have:
0.5P = P(1 - 0.15)^t
To solve for t, we can take the logarithm of both sides of the equation. The most commonly used logarithm is the natural logarithm (ln), but you can use logarithm with any base as long as you are consistent throughout the calculation. Here, we will use the natural logarithm.
ln(0.5P) = ln(P(1 - 0.15)^t)
Now, we can use the logarithmic property that states ln(ab) = ln(a) + ln(b) to simplify the equation further:
ln(0.5) + ln(P) = ln(P) + ln(1 - 0.15)^t
Since ln(P) appears on both sides of the equation, it cancels out:
ln(0.5) = ln(1 - 0.15)^t
To isolate t, we can divide both sides of the equation by ln(1 - 0.15):
t = ln(0.5) / ln(1 - 0.15)
Using a calculator, we can evaluate this expression to find the value of t.