log(subscript2)*4 = log(subscript8)*8 + .5log(subscript4)*16 True or False. I do not know how to do this if anyone could show me the steps not the answer, but the steps that would be nice.
Forgot about the .5
"Right Side = 1 + 2 = 3
So, NOT true "
should say:
Right Side = 1 + 1 = 2
So, TRUE
To determine whether the equation log(subscript2)*4 = log(subscript8)*8 + .5log(subscript4)*16 is true or false, we need to apply logarithmic properties and solve the equation step by step. Here's how to do it:
Step 1: Simplify the equation using the properties of logarithms:
Using the logarithmic property log(subscripta)*b = b * log(subscripta) and the fact that log(subscript2)*8 = 3, we can rewrite the equation as:
4 * log(subscript2) = 8 * log(subscript8) + 0.5 * 16 * log(subscript4)
Step 2: Simplify the logarithms:
Using the logarithmic property log(subscripta)*a = 1, we can further simplify the equation as:
4 * log(subscript2) = 8 * log(subscript2^3) + 0.5 * 16 * log(subscript2^2)
Step 3: Apply the exponent rule:
We know that log(subscripta)*a^b = b, so we can rewrite the equation as:
4 * log(subscript2) = 8 * 3 + 0.5 * 16 * 2
Step 4: Perform the calculations:
Simplifying the right-hand side of the equation:
4 * log(subscript2) = 24 + 16
Step 5: Simplify further:
4 * log(subscript2) = 40
Step 6: Divide both sides by 4:
log(subscript2) = 10
Step 7: Solve for the logarithm base:
By definition, log(subscripta)b = c means that a^c = b. Therefore, we need to find the power to which 2 must be raised to equal 10.
Step 8: Evaluate the base-2 logarithm:
Using basic exponentiation, we find that 2^10 = 1024. Hence, log(subscript2)10 = 10 is false.
Therefore, the statement log(subscript2)*4 = log(subscript8)*8 + .5log(subscript4)*16 is false.
here is a good summary of the log rules you should know
http://mathsfirst.massey.ac.nz/Algebra/Logarithms/RuleLog.htm
your equation:
log2 4 = log8 8 + .5log4 16
Left Side = 2
Right Side = 1 + 2 = 3
So, NOT true