a. Log8x+log(x-20)=3
To solve the equation log8x + log(x-20) = 3, we can use the properties of logarithms.
First, we can combine the two logarithms using the product rule: log(a) + log(b) = log(a * b).
So, applying this property to the equation, we have log[(8x) * (x-20)] = 3.
Next, using the definition of logarithms, we can rewrite this equation as an exponential equation: 10^3 = (8x) * (x-20).
Simplifying the exponent, we get 1000 = (8x) * (x-20).
Now, we can solve this quadratic equation by setting it equal to zero and factoring or using the quadratic formula.
Setting 1000 - (8x) * (x-20) = 0, we can factor 8x(x-20) - 1000 = 0.
Factoring further, we have 8x(x-20) - 1000 = 0.
Expanding the equation, we get 8x^2 - 160x - 1000 = 0.
Now, we can either factor this quadratic equation or use the quadratic formula to solve for x.
why didn't you finish it ??
8x^2 - 160x - 1000 = 0
x^2 - 20x - 125 = 0
(x-25)(x+5) = 0
x = 25 or x = -5
but in the original equation, x > 0
so x = 25
Apologies for not completing the solution earlier. You are correct in factoring the quadratic equation as:
8x^2 - 160x - 1000 = 0
x^2 - 20x - 125 = 0
(x-25)(x+5) = 0
From here, we obtain two possible solutions: x = 25 or x = -5.
However, when considering the original equation log8x + log(x-20) = 3, we can deduce that x must be greater than 20 (due to the presence of log(x-20)). Additionally, x must be greater than 0 since logarithms are not defined for negative numbers or zero.
Therefore, we discard x = -5 as a valid solution, leaving us with x = 25 as the final solution to the equation log8x + log(x-20) = 3.
To solve the equation log8x + log(x-20) = 3, we can use the properties of logarithms.
Step 1: Combine the logarithms using the product rule: log(A) + log(B) = log(A*B).
log8x + log(x-20) = log((8x)(x-20))
Step 2: Apply the power rule of logarithms to eliminate the logarithm with the base 8.
log((8x)(x-20)) = log(8) + log(x) + log(x-20)
Step 3: Simplify the right side using properties of logarithms.
log((8x)(x-20)) = 3log(10) + log(x) + log(x-20)
Step 4: Recall that log(10) = 1, so we can simplify further.
log((8x)(x-20)) = 3 + log(x) + log(x-20)
Step 5: Set the inside of the logarithm equal to the exponent.
(8x)(x-20) = 10^(3 + log(x) + log(x-20))
Step 6: Simplify the right side by using the rules of exponents.
(8x)(x-20) = 10^3 * 10^(log(x)) * 10^(log(x-20))
Step 7: Recall that log base 10 of a number is equivalent to the number itself.
(8x)(x-20) = 10^3 * x * (x-20)
Step 8: Simplify the equation further.
8x(x-20) = 1000x(x-20)
Step 9: Expand and rearrange the equation.
8x^2 - 160x = 1000x^2 - 20000x
Step 10: Combine like terms on both sides.
0 = 992x^2 - 19840x
Step 11: Factor out an x.
0 = x(992x - 19840)
Step 12: Set each factor equal to zero to solve for x.
x = 0 or 992x - 19840 = 0
Step 13: Solve the second equation for x.
992x - 19840 = 0
992x = 19840
x = 19840/992
x = 20
Therefore, the solution to the equation log8x + log(x-20) = 3 is x = 0 or x = 20.