When solving exponential equations using logarithms. How would you sold one that has a fraction as the exponent such as:
x^(1/4)-81=0
To solve an exponential equation with a fraction as the exponent, you can use logarithms. Specifically, you can use the logarithm with the same base as the exponential equation to eliminate the exponent and convert it into a regular algebraic equation.
Let's solve the equation you provided: x^(1/4) - 81 = 0
Step 1: Add 81 to both sides of the equation to isolate the x term:
x^(1/4) = 81
Step 2: Take the logarithm of both sides of the equation. You can choose any base for the logarithm, but it is common to use natural logarithm (ln) or base-10 logarithm (log):
log(x^(1/4)) = log(81)
Step 3: Use the logarithmic property to move the exponent down in front of the logarithm:
(1/4) * log(x) = log(81)
Step 4: Simplify the equation by evaluating the logarithms on the right side:
(1/4) * log(x) = log(81)
(1/4) * log(x) = log(3^4) [81 can be written as 3^4]
Step 5: Apply the power rule for logarithms:
(1/4) * log(x) = 4 * log(3)
Step 6: Multiply both sides of the equation by 4 to eliminate the fraction:
log(x) = 16 * log(3)
Step 7: Use the property of logarithms that states log a^n = n * log a:
log(x) = log(3^16)
Step 8: Set the arguments of both logarithms equal to each other:
x = 3^16
Step 9: Simplify the equation by evaluating the exponential expression:
x = 43,046,721
Therefore, the solution to the exponential equation x^(1/4) - 81 = 0 is x = 43,046,721.