Solve:

Log5(x-2) + log8(x-4)=
log6(x-1)

If those are logs to the same base, then

5(x-2)*8(x-4) = 6(x-1)
x = (123+√2089)/40

If those are logs to different bases (5,8,6), it takes some more work. Changing all to natural logs, we have

Log5(x-2) + log8(x-4)=
log6(x-1)

ln(x-2)/ln5 + ln(x-4)/ln8 = ln(x-1)/ln6

ln8*ln6*ln(x-2) + ln5*ln6*ln(x-4) = ln5*ln8*ln(x-1)

x ≈ 5.1797

Other than a graphical or numerical method, I don't see how to arrive at a solution.

To solve this equation, we need to apply logarithm properties and simplify the equation. Here are the steps:

Step 1: Combine the logarithms on the left side of the equation using the product rule of logarithms. The product rule states that the logarithm of a product is equal to the sum of the logarithms of its factors.

Log5(x-2) + log8(x-4) = log6(x-1)

Step 2: Apply the logarithm property to change the log base.

log_x(y) = log_z(y) / log_z(x)

Using this property, we can rewrite the equation as:

log(x-2) / log5 + log(x-4) / log8 = log(x-1) / log6

Step 3: Convert the logarithms into a common base. In this case, we can convert all logarithms to base 10 for simplicity.

log(x-2) / log5 = log(x-4) / log8 = log(x-1) / log6

Step 4: Solve for x.

We will consider each fraction separately:

log(x-2) / log5 = log(x-4) / log8

Cross-multiplying, we get:

log(x-2) * log8 = log5 * log(x-4)

Step 5: Simplify the equation further.

Using the property log_a(b) * log_b(c) = log_a(c), the equation becomes:

log(x-2) = (log5 / log8) * log(x-4)

Step 6: Solve for x.

Now that we have a single logarithm on both sides of the equation, we can cancel out the logarithms by exponentiating both sides.

Let's take the exponent of base 10:

10^(log(x-2)) = 10^[(log5 / log8) * log(x-4)]

Simplifying,

x - 2 = [(x-4)^(log5 / log8)]

Finally, solve for x by isolating it on one side of the equation:

x = 2 + [(x-4)^(log5 / log8)]

Note that the value of x will depend on the exact values of log5 and log8, which are constants that represent the logarithms of 5 and 8 in base 10.