To solve the equation [tex]\(\log 3^{x+4} = \log 229\)[/tex], we can follow these steps:
1. Understand the logarithmic property:
We know that [tex]\(\log a = \log b\)[/tex] if and only if [tex]\(a = b\)[/tex]. This means that if the logarithms of two expressions are equal, then the expressions themselves must be equal.
2. Remove the logarithms:
Given [tex]\(\log 3^{x+4} = \log 229\)[/tex], we can equate the expressions inside the logarithms:
[tex]\[
3^{x+4} = 229
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], we need to rewrite the equation. We know that the exponent of a base can be solved using the logarithmic function. Take the logarithm of both sides of the equation using base 3:
[tex]\[
\log_3 (3^{x+4}) = \log_3 (229)
\][/tex]
Using the property of logarithms [tex]\(\log_b (b^y) = y\)[/tex]:
[tex]\[
x + 4 = \log_3 (229)
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
Now, we solve for [tex]\(x\)[/tex] by isolating it:
[tex]\[
x = \log_3 (229) - 4
\][/tex]
5. Express the answer:
Finally, [tex]\(x\)[/tex] can be written as:
[tex]\[
x = \log_3 (229) - 4
\][/tex]
Using the change of base formula for logarithms, [tex]\(\log_3 (229)\)[/tex] can be converted to a common logarithm (e.g., base 10) if needed:
[tex]\[
\log_3 (229) = \frac{\log(229)}{\log(3)}
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] can be expressed as:
[tex]\[
x = \frac{\log(229)}{\log(3)} - 4
\][/tex]
So, the value of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{-4 + \frac{\log(229)}{\log(3)}}
\][/tex]
