Certainly! Let's solve the equation [tex]\( \frac{1}{7^{13}} = 7^x \)[/tex] step by step.
1. Understand the fraction: The left-hand side of the equation is a fraction which involves a power of 7. We have [tex]\( \frac{1}{7^{13}} \)[/tex].
2. Rewrite the fraction as a negative exponent: A useful property of exponents is that [tex]\( \frac{1}{a^b} = a^{-b} \)[/tex]. Using this property, we can rewrite [tex]\( \frac{1}{7^{13}} \)[/tex] as [tex]\( 7^{-13} \)[/tex].
So, the equation [tex]\( \frac{1}{7^{13}} = 7^x \)[/tex] can be rewritten as:
[tex]\[
7^{-13} = 7^x
\][/tex]
3. Equating the exponents: When the bases are the same, we can set the exponents equal to each other. Here, the base is 7 on both sides of the equation.
Therefore, we equate the exponents:
[tex]\[
-13 = x
\][/tex]
Hence, the value of [tex]\( x \)[/tex] is [tex]\(-13\)[/tex].
Thus, [tex]\( x = -13 \)[/tex].
