Sure, let's solve for [tex]\( x \)[/tex] in the equation [tex]\( 7^{x-10} = 25 \)[/tex].
1. Start with the given equation:
[tex]\[
7^{x-10} = 25
\][/tex]
2. To isolate [tex]\( x \)[/tex], take the logarithm base 7 of both sides:
[tex]\[
\log_7(7^{x-10}) = \log_7(25)
\][/tex]
3. Simplify the left-hand side using the property of logarithms [tex]\( \log_b(a^c) = c \cdot \log_b(a) \)[/tex]:
[tex]\[
(x-10) \cdot \log_7(7) = \log_7(25)
\][/tex]
Since [tex]\( \log_7(7) = 1 \)[/tex], this simplifies to:
[tex]\[
x - 10 = \log_7(25)
\][/tex]
4. Isolate [tex]\( x \)[/tex] by adding 10 to both sides:
[tex]\[
x = \log_7(25) + 10
\][/tex]
5. Compare this result with the given answer choices:
- a) [tex]\( x = \log_{25}(7) + 10 \)[/tex]
- b) [tex]\( x = \log_7(25) - 10 \)[/tex]
- c) [tex]\( x = \log_7(25) + 10 \)[/tex]
- d) [tex]\( x = \ln \left(\frac{25}{7}\right) + 10 \)[/tex]
- e) [tex]\( x = \log_{25}(7) - 10 \)[/tex]
- f) None of the above
The correct choice is:
c) [tex]\( x = \log_7(25) + 10 \)[/tex].
