To find the [tex]\(x\)[/tex]-intercept of the function [tex]\( g(x) = \log(x+4) \)[/tex], we need to determine the value of [tex]\( x \)[/tex] for which [tex]\( g(x) = 0 \)[/tex]. In other words, we set the equation [tex]\( g(x) \)[/tex] equal to zero and solve for [tex]\( x \)[/tex].
Here's the detailed step-by-step process of solving for the [tex]\( x \)[/tex]-intercept:
1. Set [tex]\( g(x) \)[/tex] equal to zero:
[tex]\[
g(x) = \log(x+4) = 0
\][/tex]
2. Recall the property of logarithms: When [tex]\( \log_b(y) = 0 \)[/tex], it implies that [tex]\( y = 1 \)[/tex], because any number to the power of zero is 1. Here, the base of the logarithm is assumed to be 10 (common logarithm).
3. Apply this property:
[tex]\[
\log(x+4) = 0 \implies x + 4 = 10^0
\][/tex]
4. Simplify the exponent:
[tex]\[
10^0 = 1
\][/tex]
Therefore:
[tex]\[
x + 4 = 1
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], subtract 4 from both sides of the equation:
[tex]\[
x + 4 - 4 = 1 - 4
\][/tex]
[tex]\[
x = 1 - 4
\][/tex]
[tex]\[
x = -3
\][/tex]
So, the [tex]\( x \)[/tex]-intercept of the function [tex]\( g(x) = \log(x+4) \)[/tex] is [tex]\( x = -3 \)[/tex]. This means that the graph of [tex]\( g(x) \)[/tex] will cross the [tex]\( x \)[/tex]-axis at [tex]\( x = -3 \)[/tex].
