Find the solution to each equation.
2^x=17
4log4^(x)=7
2^x=17
take log of both sides, then use log rules
x log2 = log 17
x = log17/log2 = appr 4.087
4log4^(x)=7
log 4^x = 1.75
4^x = 10^1.75
4^x = 56.23413...
repeat the same steps I used in the first one to finish it
To find the solutions to each equation, we need to isolate the variable x.
1. Equation: 2^x = 17
To solve this equation, we can take the logarithm of both sides of the equation. The choice of logarithm base depends on the context or problem requirements.
Using the natural logarithm (ln), we can rewrite the equation as:
ln(2^x) = ln(17)
Applying the logarithm rules, we can bring down the exponent x in front:
x ln(2) = ln(17)
Now we can solve for x by dividing both sides of the equation by ln(2):
x = ln(17) / ln(2)
Using a calculator or mathematical software, we can calculate the value of x:
x ≈ 4.08746
Therefore, the solution to the equation 2^x = 17 is approximately x ≈ 4.08746.
2. Equation: 4log4^(x) = 7
To solve this equation, we need to use the logarithmic properties to simplify it.
Using the logarithmic property log_a(b^c) = c log_a(b), the equation becomes:
4log4(x) = 7
Next, divide both sides of the equation by 4:
log4(x) = 7 / 4
Now, we need to convert the equation into exponential form. The base of the logarithm is 4, and the result (x) is equal to 4^(7/4):
x = 4^(7/4)
Using a calculator or mathematical software, we can calculate the value of x:
x ≈ 8
Therefore, the solution to the equation 4log4^(x) = 7 is x ≈ 8.