Understanding the Equation
The given equation is
log×5 + log6b = log192 + logb.
This equation involves logarithmic properties that we need to apply.
Step 1: Apply Logarithmic Properties
- The property of logarithms states that loga + logb = log(ab).
- Therefore, we can combine the logs on both sides of the equation.
Step 2: Simplify the Left Side
- Combining the logs on the left side:
- log(5 × 6b) = log(30b).
Step 3: Simplify the Right Side
- Similarly, combine logs on the right side:
- log(192 × b).
Step 4: Set the Logarithms Equal
- Now, we have:
- log(30b) = log(192b).
Step 5: Remove the Logarithms
- Since the logs are equal, we can set the arguments equal to each other:
- 30b = 192b.
Step 6: Solve for b
- Rearranging gives:
- 30b - 192b = 0,
- -162b = 0.
From here, we find that b must equal 0, but logarithms are undefined for zero and negative numbers.
Conclusion
- Hence, the equation is valid only for values of b that do not include 0 or negative values.
- The solution reveals constraints on b, indicating that it cannot be zero or negative for the logarithmic function to be defined.
This analysis leads us to conclude that the equation has no valid solution in the real number set for b.