To solve the problem, we are given the logarithmic equations:
[tex]\[
\log_2 x = 4
\][/tex]
[tex]\[
\log_7 y = 1
\][/tex]
We need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and then compute the sum [tex]\( x + y \)[/tex].
First, let's solve for [tex]\( x \)[/tex] in the equation [tex]\(\log_2 x = 4\)[/tex].
By definition of logarithms, [tex]\(\log_2 x = 4\)[/tex] means:
[tex]\[
x = 2^4
\][/tex]
Calculating [tex]\( 2^4 \)[/tex]:
[tex]\[
2^4 = 16
\][/tex]
So, [tex]\( x = 16 \)[/tex].
Next, let's solve for [tex]\( y \)[/tex] in the equation [tex]\(\log_7 y = 1\)[/tex].
Similarly, by the definition of logarithms, [tex]\(\log_7 y = 1\)[/tex] means:
[tex]\[
y = 7^1
\][/tex]
Calculating [tex]\( 7^1 \)[/tex]:
[tex]\[
7^1 = 7
\][/tex]
So, [tex]\( y = 7 \)[/tex].
Now that we have the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we need to find [tex]\( x + y \)[/tex].
[tex]\[
x + y = 16 + 7
\][/tex]
Adding these values together:
[tex]\[
16 + 7 = 23
\][/tex]
Therefore, the sum [tex]\( x + y \)[/tex] is:
[tex]\[
x + y = 23
\][/tex]
In conclusion, we have found that [tex]\( x = 16 \)[/tex], [tex]\( y = 7 \)[/tex], and their sum is [tex]\( x + y = 23 \)[/tex].
