To find the [tex]\( x \)[/tex]-intercept of the function [tex]\( g(x) = \log(x+4) \)[/tex], we need to determine the point where the graph of the function crosses the [tex]\( x \)[/tex]-axis. By definition, the [tex]\( x \)[/tex]-intercept occurs where the value of the function [tex]\( g(x) \)[/tex] is zero. Thus, we set [tex]\( g(x) \)[/tex] equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
g(x) = 0
\][/tex]
Substituting the given function into the equation:
[tex]\[
\log(x+4) = 0
\][/tex]
The logarithmic equation [tex]\(\log(x+4) = 0 \)[/tex] is asking us to find the value of [tex]\( x \)[/tex] for which the logarithm is zero. Recall that [tex]\(\log_b(a) = c \)[/tex] implies that [tex]\( b^c = a \)[/tex]. Applying this property to our equation (for a common logarithm with base [tex]\( 10 \)[/tex]):
[tex]\[
\log(x + 4) = 0 \implies 10^0 = x + 4
\][/tex]
Since [tex]\( 10^0 = 1 \)[/tex], the equation simplifies to:
[tex]\[
1 = x + 4
\][/tex]
Next, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] on one side of the equation:
[tex]\[
x + 4 = 1 \implies x = 1 - 4 \implies x = -3
\][/tex]
Therefore, the [tex]\( x \)[/tex]-intercept of [tex]\( g(x) = \log(x+4) \)[/tex] is at [tex]\( x = -3 \)[/tex].
In conclusion, the [tex]\( x \)[/tex]-intercept of the function [tex]\( g(x) = \log(x+4) \)[/tex] is [tex]\( x = -3 \)[/tex]. This is the point where the function crosses the [tex]\( x \)[/tex]-axis.
