Solve the following equations for x and check the solution using log.
2^6x-1 = 28
The answer is log56/6log2. Explain each step.
I will assume you meant:
2^(6x-1) = 28
take log of both sides and use rules of logs
(6x-1) log2 = log28
6x-1 = log28/log2
6x = log28/log2 + 1
x = (log28/(6log2)) + 1/6 which has the same value as log56/(6log2) using my calculator
so they must have done it a different way, let's see...
2^6x-1 = 28
multiply both sides by 2
2(2^(6x-1)) = 56
2^(6x) = 56
log it
6x log2 = log56
6x = log56/log2
x = log56/(6log2)
To solve the equation 2^(6x-1) = 28, we can use logarithms.
Step 1: Take the logarithm of both sides of the equation.
log(2^(6x-1)) = log(28)
Step 2: Apply the logarithmic property that allows us to bring the exponent down as a coefficient.
(6x-1)log(2) = log(28)
Step 3: Divide both sides of the equation by log(2) to isolate the variable, x.
6x - 1 = log(28)/log(2)
Step 4: Add 1 to both sides of the equation to get x by itself.
6x = log(28)/log(2) + 1
Step 5: Simplify the right side of the equation.
6x = log(28) + log(2)
Step 6: Combine the logarithms using the logarithmic property that states log(a) + log(b) = log(a*b).
6x = log(28*2)
Step 7: Simplify the right side further.
6x = log(56)
Step 8: Divide both sides of the equation by 6 to solve for x.
x = log(56)/6
Now, to check the solution using logarithms:
Plug the value of x, which is log(56)/6, back into the original equation:
2^(6x-1) = 28
Substituting x:
2^(6(log(56)/6)-1) = 28
Simplifying:
2^(log(56) - 1) = 28
Using the property of exponents, we can rewrite 2^(log(56) - 1) as:
2^(log(56))/2^1
Simplifying further:
56/2 = 28
The left side equals the right side, which verifies that the solution x = log(56)/6 is correct.