To solve the exponential equations you've provided, we'll analyze each one individually.
- First equation: \( 12^{x-5} = 12^{3x} \)
Since the bases are the same, we can set the exponents equal to one another:
\[ x - 5 = 3x \]
Solving for \( x \):
\[ x - 3x = 5 \] \[ -2x = 5 \] \[ x = -\frac{5}{2} \]
- Second equation: \( 5^{27} = 5^{-9x} \)
Again, since the bases are the same, we can set the exponents equal:
\[ 27 = -9x \]
Solving for \( x \):
\[ x = -\frac{27}{9} \] \[ x = -3 \]
- Third equation: \( 3^{x-4} + 2x = 3^{3(2x + 7)} \)
This one is not as straightforward. We simplify the right-hand side first:
First, rewrite \( 3^{3(2x + 7)} \):
\[ 3^{3(2x + 7)} = 3^{6x + 21} \]
Now, we set our equation:
\[ 3^{x-4} + 2x = 3^{6x + 21} \]
The left side grows significantly slower than the right side as \( x \) increases, while for negative \( x \) we need to find roots. This may suggest looking visually for solutions might help, or it can be solved with numerical methods.
However, without an exact method shown within basic algebra, this is often solved through other means (like graphing). For instance, we can do estimates or trial and error for specific values but determining an exact numerical solution would involve a more advanced technique or using approximation tools.
Thus, the solutions for each equation are:
- \( x = -\frac{5}{2} \)
- \( x = -3 \)
- The solution for the third equation isn't straightforward without advanced solving methods.
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