Simply
2log8 + log4 - log16 ÷ log 32
To simplify the expression, let's work step by step:
2log8 + log4 - log16 ÷ log 32
First, let's use the properties of logarithms to simplify the expression:
2log8 can be rewritten as log8^2, which is log64.
log16 ÷ log32 can be rewritten as log16/log32.
So the expression becomes:
log64 + log4 - log16/log32
Next, let's simplify the expression further:
log64 can be rewritten as log2^6, which is 6log2.
log4 can be rewritten as log2^2, which is 2log2.
So the expression becomes:
6log2 + 2log2 - log16/log32
Next, let's simplify the expression even further:
6log2 + 2log2 can be combined as 8log2.
So the expression becomes:
8log2 - log16/log32
Finally, let's simplify the expression completely:
log16 can be rewritten as log2^4, which is 4log2.
log32 can be rewritten as log2^5, which is 5log2.
So the expression becomes:
8log2 - 4log2/5log2
The log2 terms cancel out, giving us:
8 - 4/5
Simplifying the fraction:
8 - 4/5 = 40/5 - 4/5 = 36/5
Therefore, the simplified expression is 36/5.
To simplify the expression 2log8 + log4 - log16 ÷ log32, follow the steps below:
Step 1: Apply the logarithmic properties to simplify the expression.
Logarithmic Property 1: logₐb + logₐc = logₐ(bc)
Logarithmic Property 2: logₐb - logₐc = logₐ(b/c)
Applying property 1, combine the first two terms:
2log8 + log4 = log8^2 + log4 = log64 + log4
Step 2: Apply property 2 to combine the results from step 1 with the third term of the expression:
log64 + log4 - log16 ÷ log32 = log(64 * 4) - log16 ÷ log32 = log256 - log16 ÷ log32
Step 3: Simplify the final expression:
log256 - log16 ÷ log32 = log(256 / 16) ÷ log32 = log16 ÷ log32
Therefore, the simplified expression is log16 ÷ log32.