Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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]
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]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.