Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Sure, let's solve each equation step by step.
### a) [tex]\( 7^x = 70 \)[/tex]
To solve for [tex]\( x \)[/tex], we can use logarithms.
1. Start with the given equation:
[tex]\[ 7^x = 70 \][/tex]
2. Take the natural logarithm (or common logarithm) of both sides to make use of the properties of logarithms:
[tex]\[ \log(7^x) = \log(70) \][/tex]
3. Use the logarithm power rule, [tex]\( \log(a^b) = b \log(a) \)[/tex], to bring the exponent down:
[tex]\[ x \log(7) = \log(70) \][/tex]
4. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by [tex]\( \log(7) \)[/tex]:
[tex]\[ x = \frac{\log(70)}{\log(7)} \][/tex]
Therefore, the solution to [tex]\( 7^x = 70 \)[/tex] is:
[tex]\[ x = \frac{\log(70)}{\log(7)} \][/tex]
### b) [tex]\( 2(5^x) - 4 = 76 \)[/tex]
To solve for [tex]\( x \)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[ 2(5^x) - 4 = 76 \][/tex]
2. Add 4 to both sides to isolate the term with the exponent:
[tex]\[ 2(5^x) = 80 \][/tex]
3. Divide both sides by 2 to further isolate [tex]\( 5^x \)[/tex]:
[tex]\[ 5^x = 40 \][/tex]
4. Take the natural logarithm (or common logarithm) of both sides:
[tex]\[ \log(5^x) = \log(40) \][/tex]
5. Use the logarithm power rule, [tex]\( \log(a^b) = b \log(a) \)[/tex], to bring the exponent down:
[tex]\[ x \log(5) = \log(40) \][/tex]
6. Solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( \log(5) \)[/tex]:
[tex]\[ x = \frac{\log(40)}{\log(5)} \][/tex]
Therefore, the solution to [tex]\( 2(5^x) - 4 = 76 \)[/tex] is:
[tex]\[ x = \frac{\log(40)}{\log(5)} \][/tex]
These are the solutions for the given equations.
### a) [tex]\( 7^x = 70 \)[/tex]
To solve for [tex]\( x \)[/tex], we can use logarithms.
1. Start with the given equation:
[tex]\[ 7^x = 70 \][/tex]
2. Take the natural logarithm (or common logarithm) of both sides to make use of the properties of logarithms:
[tex]\[ \log(7^x) = \log(70) \][/tex]
3. Use the logarithm power rule, [tex]\( \log(a^b) = b \log(a) \)[/tex], to bring the exponent down:
[tex]\[ x \log(7) = \log(70) \][/tex]
4. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by [tex]\( \log(7) \)[/tex]:
[tex]\[ x = \frac{\log(70)}{\log(7)} \][/tex]
Therefore, the solution to [tex]\( 7^x = 70 \)[/tex] is:
[tex]\[ x = \frac{\log(70)}{\log(7)} \][/tex]
### b) [tex]\( 2(5^x) - 4 = 76 \)[/tex]
To solve for [tex]\( x \)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[ 2(5^x) - 4 = 76 \][/tex]
2. Add 4 to both sides to isolate the term with the exponent:
[tex]\[ 2(5^x) = 80 \][/tex]
3. Divide both sides by 2 to further isolate [tex]\( 5^x \)[/tex]:
[tex]\[ 5^x = 40 \][/tex]
4. Take the natural logarithm (or common logarithm) of both sides:
[tex]\[ \log(5^x) = \log(40) \][/tex]
5. Use the logarithm power rule, [tex]\( \log(a^b) = b \log(a) \)[/tex], to bring the exponent down:
[tex]\[ x \log(5) = \log(40) \][/tex]
6. Solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( \log(5) \)[/tex]:
[tex]\[ x = \frac{\log(40)}{\log(5)} \][/tex]
Therefore, the solution to [tex]\( 2(5^x) - 4 = 76 \)[/tex] is:
[tex]\[ x = \frac{\log(40)}{\log(5)} \][/tex]
These are the solutions for the given equations.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.