How do you solve 3 log(base 2)*(x-1)+ log(base 2)4=5
recall that n*log(a) = log(a^n)
so, using base 2, we have
3log(x-1)+log(4) = 5
log(x-1)^3 + 2 = 5
log(x-1)^3 = 3
(x-1)^3 = 2^3
x-1 = 2
x = 3
check:
3log2 + log4 = 5
3+2 = 5
another way would be
3log(x-1)+log(4) = 5
3log(x-1) + 2 = 5
3log(x-1) = 3
log(x-1) = 1
x-1 = 2^1 = 2
x = 3
To solve the equation 3 log(base 2)*(x-1) + log(base 2)4 = 5, follow these steps:
Step 1: Combine the logarithms using the properties of logarithms. The sum of logarithms can be written as a single logarithm, and the product of a constant and a logarithm can be written as an exponent:
log(base 2) [(x-1)^3] + log(base 2)4 = 5
Step 2: Apply the property of logarithms that states: "The sum of logarithms with the same base is equal to the logarithm of the product of the individual values":
log(base 2) [(x-1)^3 * 4] = 5
Step 3: Rewrite the equation using exponential form. In general, log(base b) y = x can be rewritten as b^x = y. Applying this to the equation:
2^5 = (x-1)^3 * 4
Step 4: Simplify the equation:
32 = 4(x-1)^3
Step 5: Divide both sides of the equation by 4:
8 = (x-1)^3
Step 6: Take the cube root of both sides of the equation:
∛(8) = ∛[(x-1)^3]
2 = x - 1
Step 7: Add 1 to both sides of the equation:
2 + 1 = x - 1 + 1
3 = x
So, the solution to the equation 3 log(base 2)*(x-1) + log(base 2)4 = 5 is x = 3.
To solve the equation 3 log(base 2)*(x-1) + log(base 2)4 = 5, we can use the properties of logarithms to simplify the equation.
Step 1: Start by applying the logarithmic properties. We know that log(base a) c + log(base a) d = log(base a) (c * d). Therefore, we can rewrite the equation as:
log(base 2) [(x - 1)^3] + log(base 2)4 = 5
Step 2: Combine the two logarithms into a single logarithm by applying the property log(base a) b + log(base a) c = log(base a) (b * c):
log(base 2) [(4 * (x - 1)^3)] = 5
Step 3: Rewrite the equation in exponential form: If log(base a) b = c, then a^c = b. Therefore, we have:
2^5 = 4 * (x - 1)^3
Step 4: Simplify the exponential equation:
32 = 4 * (x - 1)^3
Step 5: Divide both sides of the equation by 4 to isolate the cubic term:
8 = (x - 1)^3
Step 6: Take the cube root of both sides of the equation:
∛8 = ∛[(x - 1)^3]
Simplifying further:
2 = x - 1
Step 7: Add 1 to both sides of the equation:
2 + 1 = x
x = 3
Therefore, the solution to the equation 3 log(base 2)*(x-1) + log(base 2)4 = 5 is x = 3.