To determine the least integer value of [tex]\( x \)[/tex] that satisfies the inequality [tex]\( \log_3(8x - 37) > 5 \)[/tex], we need to solve the inequality step-by-step.
1. Begin by rewriting the logarithmic inequality in exponential form. Recall that if [tex]\( \log_b(a) = c \)[/tex], then [tex]\( a = b^c \)[/tex]. Thus, the inequality [tex]\( \log_3(8x - 37) > 5 \)[/tex] can be converted as follows:
[tex]\[
8x - 37 > 3^5
\][/tex]
2. Calculate [tex]\( 3^5 \)[/tex]:
[tex]\[
3^5 = 243
\][/tex]
3. Substitute [tex]\( 243 \)[/tex] back into the inequality:
[tex]\[
8x - 37 > 243
\][/tex]
4. Add 37 to both sides of the inequality to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
8x > 243 + 37
\][/tex]
5. Simplify the expression on the right-hand side:
[tex]\[
8x > 280
\][/tex]
6. Divide both sides of the inequality by 8 to solve for [tex]\( x \)[/tex]:
[tex]\[
x > \frac{280}{8}
\][/tex]
7. Simplify the fraction:
[tex]\[
x > 35
\][/tex]
The inequality [tex]\( x > 35 \)[/tex] tells us that [tex]\( x \)[/tex] must be greater than 35. The smallest integer greater than 35 is 36.
Therefore, the least integer value for which the inequality [tex]\( \log_3(8x - 37) > 5 \)[/tex] holds true is:
[tex]\[
\boxed{36}
\][/tex]
