solve for x:
x=9^(logbase9(5)+logbase9(9))
Note: Your answer must be exact and in simplest form
To solve for x, we can simplify the expression step by step.
Given: x = 9^(logbase9(5) + logbase9(9))
Step 1: Simplify the expression inside the parentheses.
logbase9(9) equals 1, since any number raised to the power of 1 is the number itself.
So, x = 9^(logbase9(5) + 1)
Step 2: Simplify further.
Using the logarithmic identities, logbasea(b) + logbasea(c) = logbasea(b * c).
x = 9^(logbase9(5 * 9))
Step 3: Simplify the product inside the logarithm.
5 * 9 equals 45.
x = 9^(logbase9(45))
Step 4: Use the change of base formula.
Since the logarithm's base is 9, we can rewrite the expression using a different base, such as 10.
logbase9(45) = logbase10(45) / logbase10(9)
Step 5: Evaluate the logarithms.
Using a calculator, we find that logbase10(45) ≈ 1.6532 and logbase10(9) ≈ 0.9542.
x = 9^(1.6532 / 0.9542)
Step 6: Simplify further.
Dividing exponents is equivalent to subtracting the logarithms with the same base.
x = 9^1.7317
Step 7: Evaluate the final exponent.
Using a calculator, we find that 9^1.7317 ≈ 47.6843
Therefore, the exact and simplest form of x is approximately x = 47.6843.