The mass of a particular substance is known to grow exponentially at a rate of 0.5% per hour. Its initial mass was 25 grams and, after t hours, it weighed 31 grams.
The equation modelling this growth is
25×1.005 t = 31.
Use the method of taking logs to solve this equation for t, giving your answer correct to the nearest hour.
Your equation is incorrect, it should say
25(1.005)^t = 31
then
1.005^t = 1.24
log (1.005^t) = log 1.24
t(log 1.005) = log 1.24
t = log 1.24/log 1.005 = appr. 43.1 hrs
Oh sorry! Thanks
To solve the equation 25 * 1.005^t = 31, we can take the logarithm (log) of both sides of the equation to eliminate the exponential term.
Taking the logarithm (log) of both sides:
log (25 * 1.005^t) = log (31)
Next, we can use the logarithmic properties to simplify the equation. Using the property log(A * B) = log(A) + log(B), we can rewrite the equation as:
log (25) + log (1.005^t) = log (31)
Since log (1.005^t) = t * log (1.005), we can further simplify the equation as:
log (25) + t * log (1.005) = log (31)
Now, we can isolate the variable t by subtracting log (25) from both sides:
t * log (1.005) = log (31) - log (25)
Using the property log(A) - log(B) = log(A/B), we can rewrite the equation as:
t * log (1.005) = log (31/25)
Finally, we can solve for t by dividing both sides of the equation by log (1.005):
t = (log (31/25)) / log (1.005)
Calculating the value of t using a calculator, we get:
t ≈ 21.8
Rounding the value to the nearest whole number, t is approximately 22 hours.
To solve the equation 25×1.005^t = 31 using the method of taking logarithms, you can follow these steps:
Step 1: Rewrite the equation in a logarithmic form.
Taking the logarithm (base 10) of both sides of the equation, we have:
log(25×1.005^t) = log(31)
Using the rule for the logarithm of a product, we can rewrite the equation as:
log(25) + log(1.005^t) = log(31)
Step 2: Simplify the logarithmic expression.
Since log(1.005^t) can be rewritten as t * log(1.005), we can simplify the equation further:
log(25) + t * log(1.005) = log(31)
Step 3: Solve for t.
Isolate t by subtracting log(25) and dividing by log(1.005):
t * log(1.005) = log(31) - log(25)
t = (log(31) - log(25)) / log(1.005)
Step 4: Calculate the result.
Plug in the values into a calculator to get an approximate value for t:
t ≈ (1.491 - 1.397) / 0.000434
t ≈ 94.18
Rounding to the nearest hour, t is approximately 94 hours.