To solve the given equation [tex]\( 8^{7x - 5} = 8^{4x - 3} \)[/tex], follow these steps:
1. Recognize the properties of exponents:
Since the bases are the same (both sides have base 8), we can set the exponents equal to each other.
2. Set up the equation:
[tex]\[
7x - 5 = 4x - 3
\][/tex]
3. Rearrange the equation:
Subtract [tex]\( 4x \)[/tex] from both sides to isolate [tex]\( x \)[/tex] on one side:
[tex]\[
7x - 4x - 5 = -3
\][/tex]
4. Simplify:
Combine like terms on the left-hand side:
[tex]\[
3x - 5 = -3
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Add 5 to both sides to get:
[tex]\[
3x = 2
\][/tex]
6. Divide by 3:
[tex]\[
x = \frac{2}{3}
\][/tex]
Hence, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = 0.6666666666666666
\][/tex]
