A new automobile costs $32,000. The value of the same automobile after t years is given by V = 32000(0.8)t. The following steps determine how many years it will take for the value to decrease to one-eighth the initial value, which is $4,000.

Question

A new automobile costs $32,000. The value of the same automobile after t years is given by V = 32000(0.8)t. The following steps determine how many years it will take for the value to decrease to one-eighth the initial value, which is $4,000.

1)weneedtocompute(0.8)t=1/8
,sowetakelogarithmofbase10onboth side: write out the result.

2) using the fact that log102 = 0.3010, compute log108. [hint: 23 = 8]

3) use the result of log108, compute log10 (1/8). [hint: 1/8 = 8-1]

4) use the result of log108, compute log100.8 [hint: 0.8 = 8/10]

5) use the result of 3) and 4), solve for t in 1)

Answer 1

To find the number of years (t) it will take for the value of the automobile to decrease to one-eighth the initial value ($4,000), we can follow the given steps:

1) The equation given is V = 32000(0.8)^t, and we need to solve for t when the value is one-eighth the initial value. In other words, we need to find t when (0.8)^t = 1/8.

To solve this, we can take the logarithm (base 10) of both sides of the equation:
log(0.8)^t = log(1/8)

Applying logarithm rules, we can bring the exponent t down:
t*log(0.8) = log(1/8)

2) Now, we need to calculate log(0.8) using the fact that log(base 10) 2 = 0.3010.

Since (0.8) can be expressed as (2/10)^1, we can rewrite it as:
log(0.8) = log((2/10)^1)

Using the logarithm rule, we bring the exponent 1 down:
log(0.8) = 1 * log(2/10)

Since log(2/10) = log(2) - log(10) = log(2) - 1, we have:
log(0.8) = 1 * (log(2) - 1) = log(2) - 1

3) Next, we need to compute log(1/8). Since (1/8) is equal to (8)^(-1), we can write it as:
log(1/8) = log((8)^(-1))

Using the logarithm rule, we bring the exponent -1 down and reverse the sign:
log(1/8) = -1 * log(8)

Since log(8) = 3 (as 2^3 = 8), we have:
log(1/8) = -1 * 3 = -3

4) Now, we have to compute log(0.8). Since 0.8 is equal to (8/10), we can write it as:
log(0.8) = log((8/10))

Using the logarithm rule, we can split the two fractions:
log(0.8) = log(8) - log(10)

Since log(8) = 3 and log(10) = 1, we have:
log(0.8) = 3 - 1 = 2

5) Finally, we can use the results from steps 3) and 4) to solve for t in step 1).

Substituting the values into the equation derived in step 1):
t * (log(2) - 1) = -3

Now, let's rearrange the equation to solve for t:
t = -3 / (log(2) - 1)

Calculating the value on the right-hand side:
t = -3 / (0.3010 - 1)
t = -3 / (-0.6990)
t ≈ 4.29 years

Therefore, it will take approximately 4.29 years for the value of the automobile to decrease to one-eighth the initial value ($4,000).