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.