Solve the logarithm equation. Express the solution in exact form only.
log[6](64x^3+1)-log[6](4x+1)=1
log[6] ( (64x^3 + 1)/(4x+1) ) = 1
(64x^3+1)/(4x+1) = 6^1
I can reduce the left side ...
(4x+1)(16x^2 -4x+1)/(4x+1) = 6
16x^2 - 4x - 5 = 0
x = (4 ± √336)/32
but in log (4x+1) , 4x+1 > 0
x > -1/4, so we have to reject the negative x value
x = (4 + √336)/32
= (4 + 4√21)/32
= (1 + √21)/8
To solve the logarithm equation log[6](64x^3+1) - log[6](4x+1) = 1, we can use the properties of logarithms to simplify the equation and find the solution.
The first property we can use is the quotient rule of logarithms, which states that log[a](b) - log[a](c) = log[a](b/c). Using this property, we can rewrite the equation as a single logarithm:
log[6]((64x^3+1)/(4x+1)) = 1
Next, let's rewrite 1 as a logarithm with base 6. Since log[a](1) = 0 for any base a, we have:
log[6]((64x^3+1)/(4x+1)) = log[6](6^0)
Now, according to the property of logarithms that states log[a](b^c) = c * log[a](b), we can rewrite the equation as:
log[6]((64x^3+1)/(4x+1)) = 0
Since the logarithm equation states that log[a](b) = c if and only if a^c = b, we can raise both sides of the equation to the power of 6:
(64x^3+1)/(4x+1) = 6^0
Simplifying further:
(64x^3+1)/(4x+1) = 1
To eliminate the fraction, we can multiply both sides of the equation by (4x+1):
(4x+1)*((64x^3+1)/(4x+1)) = (4x+1)*1
Canceling out (4x+1) on the left side gives:
64x^3 + 1 = 4x + 1
Subtracting 4x + 1 from both sides:
64x^3 - 4x = 0
Now, we can factor out x:
x(64x^2 - 4) = 0
Setting each factor equal to zero:
x = 0
64x^2 - 4 = 0
For the second equation, we can solve for x by dividing both sides by 64:
64x^2 = 4
x^2 = 4/64
x^2 = 1/16
Taking the square root of both sides:
x = ±√(1/16)
x = ±1/4
Therefore, the solutions to the logarithm equation log[6](64x^3+1) - log[6](4x+1) = 1 are x = 0, x = 1/4, and x = -1/4.