The product of all solutions of the equation e5(logex)2+3=x8, x>0, is:
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Step-by-Step Solution
Step 1: Take natural logarithm on both sides
To solve an equation where the variable is in the exponent, we can take the logarithm of both sides. Since the base of the exponential is e, we use the natural logarithm (base e). This allows us to bring down the exponent using the logarithm property logb(ac)=clogba and logeek=k.
Step 2: Rearrange into a quadratic equation
Let y=logex. Substituting y into the equation transforms it into a standard quadratic form ay2+by+c=0. This makes it easier to solve for y.
Step 3: Solve the quadratic equation for logex
We solve the quadratic equation 5y2−8y+3=0 by factoring. The factors are (5y−3) and (y−1), which gives us two possible values for y: 3/5 and 1.
Step 4: Solve for x
Now we substitute back y=logex and solve for x. Using the definition of logarithm, if logba=c, then a=bc. This gives us two solutions for x. Both solutions are positive, satisfying the condition x>0.
Step 5: Calculate the product of solutions
The problem asks for the product of all solutions. We multiply the two values of x we found. When multiplying exponential terms with the same base, we add their exponents.