Certainly! We are tasked with determining whether the logarithm of 225 to the base 6 is greater than a particular value [tex]\( x \)[/tex]. Specifically, we are given that:
[tex]\[
\log_6 225 > x
\][/tex]
We need to find out what value [tex]\( x \ should be such that this inequality holds true. To find \(\log_6 225\)[/tex], we start by understanding the logarithmic function [tex]\(\log_b a\)[/tex], which represents the exponent [tex]\( y \)[/tex] such that [tex]\( b^y = a \)[/tex].
Here, [tex]\( b = 6 \)[/tex] and [tex]\( a = 225 \)[/tex]. We are essentially looking for:
[tex]\[
6^y = 225
\][/tex]
By solving this, we find that:
[tex]\[
y = \log_6 225
\][/tex]
To verify the inequality, we calculate the value of [tex]\(\log_6 225\)[/tex]. Upon determining the value, we find:
[tex]\[
y \approx 3.0227831889387713
\][/tex]
Thus, the inequality [tex]\(\log_6 225 > x\)[/tex] can be satisfied by [tex]\( x \)[/tex] values less than this logarithm value:
[tex]\[
3.0227831889387713 > x
\][/tex]
In conclusion, any [tex]\( x \)[/tex] less than [tex]\( 3.0227831889387713 \)[/tex] will satisfy the inequality [tex]\(\log_6 225 > x\)[/tex]. Therefore:
[tex]\[
\log_6 225 \approx 3.0227831889387713
\][/tex]
So, the solution to the inequality [tex]\( \log_6 225 > x \)[/tex] implies that [tex]\( x\)[/tex] must be less than approximately 3.0227831889387713.
