solve the exponential equation Express in exact form.
5e^xlog_9 = 11
Is x = (ln11-ln5/log_9 exact form??????
assuming you mean
5e^(xlog 9) = 11
then
e^(x ln 9) = 11/5
x ln 9 = ln(11/5)
x = (ln 11 - ln 5)/ln 9
you are correct
To solve the exponential equation 5e^xlog_9 = 11 and express the answer in exact form, we need to isolate the variable x.
1. Start by dividing both sides of the equation by 5:
e^xlog_9 = 11/5
2. We can rewrite e^xlog_9 as (9^(log_9))(e^x), since log_9 is the logarithm base. This property allows us to express e^xlog_9 as a power of the base 9.
3. Applying the aforementioned property, we get:
9^(log_9)(e^x) = 11/5
4. The base 9 and the logarithm base 9 cancel each other out, leaving us with:
e^x = 11/5
5. Now, to solve for x, we take the natural logarithm (ln) of both sides:
ln(e^x) = ln(11/5)
The natural logarithm undoes the effect of the exponential function.
6. By using the property ln(e^x) = x, we can simplify the equation further:
x = ln(11/5)
Therefore, x = ln(11/5) is the exact expression for the solution to the equation.
Regarding your second question, if you mean to express ln(11/5) in terms of logarithms with base 9 (log_9), it is not directly possible since ln represents the natural logarithm (base e) while log_9 represents the logarithm with base 9. However, you can use the change of base formula to convert ln(11/5) into log_9.