use properties of logarithms to find the exact value of the expression. sqrtlog81 3
To find the exact value of the expression sqrt(log81 3), we can use the properties of logarithms, particularly the change of base formula.
Step 1: Start by expressing the expression in terms of a common base. Since the square root can be expressed as a fractional exponent, we can rewrite sqrt(log81 3) as (log81 3)^(1/2).
Step 2: Now, we will rewrite the logarithm with a different base. A common base to choose is 3, as we are taking the logarithm base 3 of 81. Applying the change of base formula, we have:
log81 3 = log3 3 / log3 81
Step 3: Simplify the expression:
log81 3 = 1 / (log3 81)
Step 4: Now, we need to simplify the denominator further. Rewrite the logarithm base 3 of 81 as a power of 3:
log3 81 = log3 (3^4)
Using the logarithmic property, we can bring down the exponent 4 as the coefficient of the logarithm:
log3 81 = 4 * log3 3
Step 5: Substitute this simplified expression back into the denominator:
log81 3 = 1 / (4 * log3 3)
Step 6: Finally, substitute this result back into the original expression:
(sqrt(log81 3)) = (log81 3)^(1/2) = [(1 / (4 * log3 3))]^(1/2)
Now, you can simplify this or evaluate it further using a calculator.