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.
