Solve the logarithmic equation. Express your solutions in exact form only. Please show all of your work.

log base 6 (64x^3+1)-log base 6 (4x+1)=1

log 6[ (64x^3+1)/(4x+1) ] = 1

(64x^3+1)/(4x+1) = 6^1
(4x+1)(16x^2 - 4x + 1)/(4x+1) = 6
16x^2 - 4x - 5 = 0
x= (4 ± √336)/32
= (4 ± 4√21)/32
= (1 ± 21)/8

We have to reject the negative answer or else the log would be undefined
so
x = (1 + √21)/8

First, we can simplify the equation by using the logarithmic property of subtraction:

log₆(64x³ + 1) / log₆(4x + 1) = 1

Next, we can convert the logarithmic equation into exponential form:

6¹ = 64x³ + 1 / (4x + 1)

Simplifying further:

6 = 64x³ + 1 / (4x + 1)

Multiply both sides of the equation by (4x + 1) to eliminate the denominator:

6(4x + 1) = 64x³ + 1

24x + 6 = 64x³ + 1

Rearrange the equation:

64x³ - 24x + 5 = 0

Now, we have a polynomial equation. Unfortunately, there is no simple algebraic method to solve a cubic equation like this one. However, we can use numerical or approximation methods (like Newton's method) to find an approximate solution.

If you are looking for an exact solution, you might need to use techniques from calculus or other advanced mathematical methods.