To solve the equation [tex]\( 3^{5x} = 3^{x-7} \)[/tex] for [tex]\( x \)[/tex], let's follow a step-by-step approach.
1. Recognize the form of the equation: The given equation is an exponential equation where the bases on both sides are the same. When the bases are the same, we can equate the exponents directly. This is because if [tex]\( a^m = a^n \)[/tex] and [tex]\( a \neq 0,1 \)[/tex], then [tex]\( m = n \)[/tex].
2. Set the exponents equal to each other:
[tex]\[
5x = x - 7
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
- First, subtract [tex]\( x \)[/tex] from both sides to isolate the variable terms on one side of the equation.
[tex]\[
5x - x = -7
\][/tex]
- Simplify the left-hand side:
[tex]\[
4x = -7
\][/tex]
- Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{7}{4}
\][/tex]
Next, let's consider other potential solutions involving complex numbers, because in exponential equations, it is possible to have complex solutions due to the periodic nature of the exponential function.
4. Considering complex solutions:
- The general form for a complex logarithm solution involves including [tex]\( 2k\pi i \)[/tex], where [tex]\( k \)[/tex] is an integer, to account for the periodicity of the exponential function.
Using this knowledge and algebraic manipulations, we arrive at the complex solutions which include terms involving [tex]\( \pi i \)[/tex] and the natural logarithm function. These take the forms:
[tex]\[
x = \frac{(-\log 2187 - 2k\pi i)}{4\log 3}
\][/tex]
For specific values of [tex]\( k \)[/tex], we have the following solutions:
When [tex]\( k = 0 \)[/tex]:
[tex]\[
x = \frac{-\log 2187}{4\log 3}
\][/tex]
When [tex]\( k = 1 \)[/tex]:
[tex]\[
x = \frac{-\log 2187 - 2\pi i}{4\log 3}
\][/tex]
When [tex]\( k = -1 \)[/tex]:
[tex]\[
x = \frac{-\log 2187 + 2\pi i}{4\log 3}
\][/tex]
Additionally, we observe another form:
[tex]\[
x = -\frac{7}{4} + \frac{\pi i}{\log 3}
\][/tex]
5. Compile all solutions:
- The real solution is:
[tex]\[
x = -\frac{7}{4}
\][/tex]
- The complex solutions include:
[tex]\[
x = \frac{-\log 2187 - 2\pi i}{4\log 3}
\][/tex]
[tex]\[
x = \frac{-\log 2187 + 2\pi i}{4\log 3}
\][/tex]
[tex]\[
x = -\frac{7}{4} + \frac{\pi i}{\log 3}
\][/tex]
So, the solutions to the equation [tex]\( 3^{5x} = 3^{x-7} \)[/tex] for [tex]\( x \)[/tex] are:
[tex]\[
x = -\frac{7}{4}, \, \frac{-\log 2187 - 2k\pi i}{4\log 3} \text{ for } k = 1, \text{ and } k = -1, \, \text{and another form } -\frac{7}{4} + \frac{\pi i}{\log 3}.
\][/tex]
