Question

If the price of gas started at $2.56 per gallon and increased at a rate of 4% per year, after how many years will the price of gasoline per gallon reach or exceed $5?

The price of gas is modeled by f(x) = 2.56(1.04)x.
Use logarithms to find the answer to the question.

Answer 1

Answer: 18 years

Work Shown:

`f(\text{x}) = 2.56(1.04)^\text{x}\\\\5 = 2.56(1.04)^\text{x}\\\\5/2.56 = (1.04)^\text{x}\\\\1.953125 = (1.04)^\text{x}\\\\\log(1.953125) = \log(1.04^\text{x})\\\\\log(1.953125) = \text{x}*\log(1.04)\\\\\text{x} = (\log(1.953125))/(\log(1.04))\\\\\text{x} \approx 17.0682937693249\\\\`

The steps above show f(x) replaced with 5. Then you'd use logarithms to isolate the variable x. The relevant useful log rule is
`\log(A^B) = B\log(A)` so we can pull down the exponent.

From here it seems your teacher wants you to round up to the nearest integer.

If we plugged in x = 17, then f(x) = 4.99 which is one cent too small.

While x = 18 leads to f(x) = 5.19

Therefore, x = 18 has the price reach or exceed $5