solve it:

a) 3 log x with base 8 + 2 = 0
b)2 log (x-1)with base 6 =log 4 with base 6

a)

3 log8 x = -2
log8 x = -2/3
8^(-2/3) = x
x = 1/4

b) 2 log6 (x-1) = log6 4
log6 (x-1)^2 = log6 4

(x-1)^2 = 4
x-1 = ±2
x = 3 or x = -1 , but for log(x-1) to be defined, x>1

so x = 3

a) To solve the equation 3 log(x) with base 8 + 2 = 0, we can follow these steps:

Step 1: Start by isolating the logarithmic expression.
Subtracting 2 from both sides of the equation gives: 3 log(x) with base 8 = -2.

Step 2: Remove the logarithmic term by applying the exponentiation property.
Since the base of the logarithm is 8, we can rewrite the equation as: x^3 with base 8 = 8^-2.

Step 3: Simplify the right side of the equation.
The term 8^-2 can be expressed as 1/8^2 or 1/64, so the equation becomes: x^3 with base 8 = 1/64.

Step 4: Convert the equation to exponential form.
Rewrite the equation in exponential form as 8^(-3) = 1/64. The base 8 is raised to the power of -3 to give 1/64.

Step 5: Solve for x.
Since the bases are the same, we set the exponents equal to each other: -3 = -2.

Step 6: Find the value of x.
Solve the equation -3 = -2 to obtain x = 1.

Therefore, the solution to the equation 3 log(x) with base 8 + 2 = 0 is x = 1.

b) To solve the equation 2 log(x-1) with base 6 = log 4 with base 6, we can follow these steps:

Step 1: Combine the logarithmic terms.
Using the property log(a) + log(b) = log(a * b), we can rewrite the equation as log(x-1)^2 with base 6 = log 4 with base 6.

Step 2: Eliminate the logarithmic terms.
If two logarithms with the same base are equal, then their arguments must be equal. Therefore, we set (x-1)^2 = 4.

Step 3: Solve for x.
Take the square root of both sides of the equation to eliminate the exponent: x-1 = ±2.

When we solve for x with the positive square root, we get: x-1 = 2 which gives x = 3.
When we solve for x with the negative square root, we get: x-1 = -2 which gives x = -1.

Step 4: Check if the solutions are valid.
Since the logarithm is not defined for negative numbers, x = -1 is an extraneous solution. Therefore, the valid solution is x = 3.

Hence, the solution to the equation 2 log(x-1) with base 6 = log 4 with base 6 is x = 3.