Please help!!!
The mass of a particular substance is known to grow exponentially at a rate of 1.5% per day. Its initial mass was 230 grams and, after t days, it weighed 325 grams.
The equation modelling this growth is
230×1.015 t = 325.
Use the method of taking logs to solve this equation for t, giving your answer correct to the nearest day.
I think you have a typo, and meant to say ...
230(1.015)^t = 325
(your equation would have a linear growth, not exponentially)
1.015^t = 325/230 = 1.4130435
log(1.015^t) = log 1.4130435
t = log 1.4130435/log 1.015 = 23.222
so it will take appr. 23 days.
To solve the equation 230 × 1.015^t = 325 using the method of taking logs, we will first take the natural logarithm (ln) of both sides of the equation. This will allow us to isolate the exponent.
ln(230 × 1.015^t) = ln(325)
Next, we can use the logarithmic property that ln(a * b) = ln(a) + ln(b) to rewrite the left side of the equation:
ln(230) + ln(1.015^t) = ln(325)
Using the logarithmic property ln(x^y) = y * ln(x), we can further simplify the equation:
ln(230) + t * ln(1.015) = ln(325)
Now, we can isolate the variable t by subtracting ln(230) from both sides of the equation:
t * ln(1.015) = ln(325) - ln(230)
Finally, we can solve for t by dividing both sides of the equation by ln(1.015):
t = (ln(325) - ln(230)) / ln(1.015)
Using a calculator, we can evaluate this expression to find the value of t. Rounding the result to the nearest day will give us the final answer.
To solve the equation 230×1.015^t = 325 using the method of taking logs, we can isolate the exponential term and then take the logarithm of both sides.
Step 1: Isolate the exponential term
First, divide both sides of the equation by 230 to isolate the exponential term:
1.015^t = 325/230
Step 2: Take the logarithm of both sides
Now, take the logarithm of both sides of the equation. You can use any logarithm base, but commonly used bases are 10 (log) and e (ln).
Taking the natural logarithm (ln) of both sides gives:
ln(1.015^t) = ln(325/230)
Step 3: Apply the logarithmic property
Using the logarithmic property, we can bring down the exponent t in front of the logarithm:
t * ln(1.015) = ln(325/230)
Step 4: Solve for t
To solve for t, divide both sides of the equation by ln(1.015):
t = ln(325/230) / ln(1.015)
Step 5: Calculate t
Now, use a calculator to compute the value of t by plugging in the values and calculating:
t ≈ ln(325/230) / ln(1.015) ≈ 17.31
To obtain the answer in days, round t to the nearest day:
t ≈ 17.31 ≈ 17 days
Therefore, the solution to the equation is t ≈ 17 days.