Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Sure! Let's solve each of the given equations for [tex]\( x \)[/tex] step by step and round the answers to two decimal places.
### 10. [tex]\(\log_4 x = 3\)[/tex]
To solve this equation, we need to rewrite it in its exponential form:
[tex]\[ x = 4^3 \][/tex]
Now, calculate [tex]\( 4^3 \)[/tex]:
[tex]\[ 4^3 = 64 \][/tex]
So, the solution is:
[tex]\[ x = 64 \][/tex]
### 11. [tex]\(\ln x = -2.5\)[/tex]
To solve this logarithmic equation, we need to rewrite it in its exponential form using the natural logarithm property:
[tex]\[ x = e^{-2.5} \][/tex]
Calculate [tex]\( e^{-2.5} \)[/tex]:
[tex]\[ e^{-2.5} \approx 0.08 \][/tex]
So, the solution is:
[tex]\[ x \approx 0.08 \][/tex]
### 12. [tex]\(2 \cdot \log_3 x + 4 = 1\)[/tex]
First, isolate the logarithmic term:
[tex]\[ 2 \cdot \log_3 x = 1 - 4 \][/tex]
[tex]\[ 2 \cdot \log_3 x = -3 \][/tex]
Now, divide by 2:
[tex]\[ \log_3 x = -1.5 \][/tex]
Rewrite in its exponential form:
[tex]\[ x = 3^{-1.5} \][/tex]
Calculate [tex]\( 3^{-1.5} \)[/tex]:
[tex]\[ 3^{-1.5} \approx 0.19 \][/tex]
So, the solution is:
[tex]\[ x \approx 0.19 \][/tex]
### 13. [tex]\(10 \cdot \log (2x) = 30\)[/tex]
First, isolate the logarithmic term:
[tex]\[ \log (2x) = \frac{30}{10} \][/tex]
[tex]\[ \log (2x) = 3 \][/tex]
Rewrite it in its exponential form:
[tex]\[ 2x = 10^3 \][/tex]
Calculate [tex]\( 10^3 \)[/tex]:
[tex]\[ 10^3 = 1000 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = 1000 \implies x = \frac{1000}{2} = 500 \][/tex]
So, the solution is:
[tex]\[ x = 500.00 \][/tex]
### Summary
The solutions to the equations are:
10. [tex]\( x = 64 \)[/tex]
11. [tex]\( x \approx 0.08 \)[/tex]
12. [tex]\( x \approx 0.19 \)[/tex]
13. [tex]\( x = 500.00 \)[/tex]
### 10. [tex]\(\log_4 x = 3\)[/tex]
To solve this equation, we need to rewrite it in its exponential form:
[tex]\[ x = 4^3 \][/tex]
Now, calculate [tex]\( 4^3 \)[/tex]:
[tex]\[ 4^3 = 64 \][/tex]
So, the solution is:
[tex]\[ x = 64 \][/tex]
### 11. [tex]\(\ln x = -2.5\)[/tex]
To solve this logarithmic equation, we need to rewrite it in its exponential form using the natural logarithm property:
[tex]\[ x = e^{-2.5} \][/tex]
Calculate [tex]\( e^{-2.5} \)[/tex]:
[tex]\[ e^{-2.5} \approx 0.08 \][/tex]
So, the solution is:
[tex]\[ x \approx 0.08 \][/tex]
### 12. [tex]\(2 \cdot \log_3 x + 4 = 1\)[/tex]
First, isolate the logarithmic term:
[tex]\[ 2 \cdot \log_3 x = 1 - 4 \][/tex]
[tex]\[ 2 \cdot \log_3 x = -3 \][/tex]
Now, divide by 2:
[tex]\[ \log_3 x = -1.5 \][/tex]
Rewrite in its exponential form:
[tex]\[ x = 3^{-1.5} \][/tex]
Calculate [tex]\( 3^{-1.5} \)[/tex]:
[tex]\[ 3^{-1.5} \approx 0.19 \][/tex]
So, the solution is:
[tex]\[ x \approx 0.19 \][/tex]
### 13. [tex]\(10 \cdot \log (2x) = 30\)[/tex]
First, isolate the logarithmic term:
[tex]\[ \log (2x) = \frac{30}{10} \][/tex]
[tex]\[ \log (2x) = 3 \][/tex]
Rewrite it in its exponential form:
[tex]\[ 2x = 10^3 \][/tex]
Calculate [tex]\( 10^3 \)[/tex]:
[tex]\[ 10^3 = 1000 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ 2x = 1000 \implies x = \frac{1000}{2} = 500 \][/tex]
So, the solution is:
[tex]\[ x = 500.00 \][/tex]
### Summary
The solutions to the equations are:
10. [tex]\( x = 64 \)[/tex]
11. [tex]\( x \approx 0.08 \)[/tex]
12. [tex]\( x \approx 0.19 \)[/tex]
13. [tex]\( x = 500.00 \)[/tex]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.