Write 4log2+log6-log3 as a single logarithm.

Explain with steps please!~~

note: to subtract log with equal base (in this case, the base is 10) we just divide the terms inside the log. thus,

4log(2) + log(6) - log(3)
4log(2) + log(6/3)
4log(2) + log(2)
finally we add:
5log(2) or
log(32)

hope this helps~ :)

Well, logarithms can be a little tricky, but don't worry, I'm here to make it fun! So, let's dive in.

First, let's take a look at the given expression: 4log2 + log6 - log3.

Step 1: Let's use some log rules to simplify this expression.
Remember that the log rule says: log(a) + log(b) = log(a * b).

Step 2: Apply the log rule and simplify.
Using the log rule mentioned above, we can rewrite the expression as:
log(2^4) + log(6) - log(3).

Step 3: Simplify further.
Remember that raising a number to a power and then taking the logarithm is like undoing each other. So, log(2^4) is equal to simply 4.
The expression now becomes:
4 + log(6) - log(3).

Step 4: Combine like terms.
We can't combine those numbers with the logarithms, so we'll just leave the expression as it is.

And Voila! The expression in a single logarithm is: 4 + log(6) - log(3).

See, logarithms are not that scary! They can actually be quite log-ical.

To write 4log2 + log6 - log3 as a single logarithm, we can use the properties of logarithms.

Step 1: Use the power rule, which states that log base b of M to the power of N is equal to N times log base b of M. Apply this rule to 4log2:
4log2 = log2(2^4) = log2(16).

Step 2: Use the product rule, which states that log base b of M times N is equal to the sum of log base b of M and log base b of N. Apply this rule to log6 - log3:
log6 - log3 = log6/3 = log2.

Step 3: Rewrite the expression using the results from step 1 and step 2:
4log2 + log6 - log3 = log2(16) + log2(6/3).

Step 4: Apply the quotient rule, which states that log base b of M/N is equal to log base b of M minus log base b of N. Apply this rule to log2(6/3):
log2(6/3) = log2(2).

Step 5: Combine the logarithms using the sum rule, which states that log base b of M plus log base b of N is equal to log base b of M times N:
log2(16) + log2(2) = log2(16 * 2).

Step 6: Simplify the expression inside the logarithm:
log2(16 * 2) = log2(32).

Step 7: The final result is log base 2 of 32.

To write the expression 4log2 + log6 - log3 as a single logarithm, we need to use logarithmic properties to combine the terms.

Step 1: Apply the power rule of logarithms to the first term 4log2. According to the rule, the coefficient of the logarithm becomes the exponent of the base.

So, 4log2 can be rewritten as log2^4.

Step 2: Combine the terms using the addition/subtraction properties of logarithms.

Now we have log2^4 + log6 - log3.

Step 3: Apply the product rule of logarithms to combine log2^4 and log6. According to the rule, the sum of two logarithms with the same base can be written as a single logarithm of their product.

Therefore, log2^4 + log6 = log(2^4 * 6).

Step 4: Simplify the expression further.

The product inside the logarithm can be simplified as follows: 2^4 * 6 = 16 * 6 = 96.

So, log2^4 + log6 can be rewritten as log96.

Step 5: Combine log96 and log3 using the subtraction property of logarithms. According to the rule, the difference of two logarithms with the same base can be written as a single logarithm of their quotient.

Therefore, log96 - log3 = log(96 / 3).

Step 6: Simplify the expression inside the logarithm: 96 / 3 = 32.

Thus, log96 - log3 can be rewritten as log32.

Hence, the expression 4log2 + log6 - log3 can be written as a single logarithm: log32.