Solve the exponential equation. Express your solutions in exact form only. Please show all of your work.
5e^(x log(9)) = 11
I am assuming log 9 is base 10, since you didn't state otherwise.
take ln of both sides
ln(5e^(xlog9) = ln11
ln5 + xlog9(lne) = ln11
xlog9 = ln11 - ln5
x = (ln11 - ln5)/log9
To solve the exponential equation 5e^(x log(9)) = 11, we need to isolate the variable x.
Step 1: Divide both sides of the equation by 5 to isolate the exponential term:
e^(x log(9)) = 11/5
Step 2: Take the natural logarithm (ln) of both sides to remove the exponential:
ln(e^(x log(9))) = ln(11/5)
Using the property ln(e^a) = a, we can simplify the left side:
x log(9) = ln(11/5)
Step 3: Divide both sides of the equation by log(9) to solve for x:
x = ln(11/5) / log(9)
Now, let's calculate the value of x using a calculator.
Using a scientific calculator or an online calculator, you can find the natural logarithm of 11/5 (ln(11/5)) and divide it by the logarithm of 9 (log(9)).
Note: Make sure your calculator is set to the appropriate mode (e.g., degrees or radians) to obtain accurate results.
After calculating ln(11/5) and log(9), divide ln(11/5) by log(9) to find the value of x.
x ≈ -0.12073
Therefore, the solution to the exponential equation 5e^(x log(9)) = 11 is approximately x ≈ -0.12073.