I am confused on how to write this function as an exponential function having a base 10:
P(t)= 50e^(0.5t).
The answer in my textbook is:
P(t) =50(10)^(t/4.6)
we want to change
e^(.5t) to 10^(kt)
let
e^(.5t) = 10^(kt)
log it
log[e^(.5t)] = log [10^(kt)]
.5t(log e) = kt(log 10)
.5 log e = k
k = .21715 or 1/4.605
so e^(.5t) = 10^(.21715t) or
e^(.5t) = 10^(t/4.605) as the have
e^.5t = 10^x
.5 t = ln (10^x)
.5 t = x ln 10
x = .5t/2.3 =t/4.6
so
P(t) = 50 (10)^(t/4.6)
To rewrite the function P(t) = 50e^(0.5t) as an exponential function with base 10, you can use the property that e^(ln(x)) = x.
First, take the natural logarithm (ln) of both sides of the equation to eliminate the exponential term e:
ln(P(t)) = ln(50e^(0.5t))
Using the property mentioned above, we can simplify the equation further:
ln(P(t)) = ln(50) + 0.5t ln(e)
Since ln(e) is equal to 1, the equation becomes:
ln(P(t)) = ln(50) + 0.5t
Now, we need to rewrite the equation using base 10 logarithm. The property that ln(x) = log10(x) / log10(e) is useful here:
ln(P(t)) = ln(50) + 0.5t
Multiplying both sides of the equation by log10(e) gives:
ln(P(t)) * log10(e) = (ln(50) + 0.5t) * log10(e)
Now, substituting log10(e) with the value 0.4343 (approximately):
0.4343 * ln(P(t)) = 0.4343 * (ln(50) + 0.5t)
Finally, rearranging the equation to isolate P(t) yields the desired exponential function:
ln(P(t)) = ln(50) + 0.2171t
Applying the property that log10(x) = ln(x) / ln(10), we can rewrite the equation as:
ln(P(t)) / ln(10) = ln(50) / ln(10) + 0.2171t / ln(10)
Simplifying the right-hand side gives:
ln(P(t)) / ln(10) = 1.69897 + 0.2171t / ln(10)
Since ln(10) is approximately 2.3026, we can further simplify the equation:
ln(P(t)) / ln(10) = 1.69897 + 0.09401t
Now, we can rewrite the equation using the common logarithm (log10):
log10(P(t)) = 1.69897 + 0.09401t
Finally, we can raise both sides of the equation to the power of 10 to obtain the exponential function with base 10:
P(t) = 10^(1.69897 + 0.09401t)
Simplifying further:
P(t) = 10^1.69897 * 10^(0.09401t)
P(t) = 50 * (10^(t/10.638))
The final form of the exponential function with base 10 is:
P(t) = 50 * (10^(t/10.638))
To rewrite the function P(t) = 50e^(0.5t) as an exponential function with base 10, we can use the property that e^x = (10^(log base 10 of e))^x.
First, we need to find the value of log base 10 of e. Using a calculator, we find that log base 10 of e is approximately 0.4343.
Now, let's substitute this value into the equation:
P(t) = 50 * (10^(log base 10 of e))^0.5t
Using the property mentioned earlier, we can rewrite it as:
P(t) = 50 * (10^(0.4343))^0.5t
Simplifying further:
P(t) = 50 * (10^(0.4343))^t^0.5
Since t^0.5 is equal to √t, we can write it as:
P(t) = 50 * (10^(0.4343))^√t
To simplify the expression (10^(0.4343)), we can evaluate it using a calculator:
(10^(0.4343)) ≈ 2.718
Now we can replace this value back into the equation:
P(t) = 50 * 2.718^√t
Therefore, the exponential function with base 10 representation of P(t)= 50e^(0.5t) is:
P(t) = 50 * (10^(0.4343))^√t
Simplifying further:
P(t) = 50 * 2.718^√t
In your textbook, the answer is given as P(t) = 50(10)^(t/4.6), which is equivalent to our answer of P(t) = 50 * 2.718^√t. The slight difference may be due to rounding the value of log base 10 of e to a decimal approximation.