Solve log_81 32x-log_81(x-3)=3/4 for x
a. x=-81/5
b. 24/31
c. no solution
We have the equation log_81 32x - log_81(x-3) = 3/4.
Using the properties of logarithms, we can rewrite this equation as log_81[(32x)/(x-3)] = 3/4.
We can now convert to exponential form to get (32x)/(x-3) = 81^(3/4).
Simplifying the right side, we have 81^(3/4) = (3^4)^(3/4) = 3^3 = 27.
Multiplying both sides of the equation by x-3, we get 32x = 27(x-3).
Expanding, we have 32x = 27x - 81.
Rearranging, we have 32x - 27x = -81.
Combining like terms, we have 5x = -81.
Dividing both sides by 5, we get x = -81/5.
Therefore, the solution to the equation log_81 32x - log_81(x-3) = 3/4 is x = -81/5.
Therefore, the correct answer is a) x = -81/5.
correct answer is c. no solution
Apologies for the mistake. Let's double-check the solution.
After obtaining the equation 5x = -81, we divide both sides by 5, which gives x = -81/5.
Now let's substitute this x value back into the original equation to verify if it holds true.
log_81 32x - log_81(x-3) = 3/4
Substituting x = -81/5:
log_81 (32(-81/5)) - log_81((-81/5)-3) = 3/4
log_81 (-67.2) - log_81 (-96.2/5) = 3/4
At this point, we can see that both values within the logarithms are negative, but logarithms are only defined for positive values. Therefore, we conclude that there is no solution to the equation.
Hence, the correct answer is indeed c) no solution.