Express as a single logarithm and simplify: log x – log (8x) + 2 log 4
log 2
Thank you, i was checking my answer.
To express the expression as a single logarithm, we can use the following properties of logarithms:
1. log(a) - log(b) = log(a/b) (quotient rule)
2. n log(a) = log(a^n) (power rule)
Now let's simplify the given expression step by step:
log x – log (8x) + 2 log 4
Using the quotient rule, we can combine the first two terms:
= log(x / (8x)) + 2 log(4)
Simplifying the first term inside the logarithm:
= log(1/8) + 2 log(4)
Using the power rule to simplify the second term:
= log(1/8) + log(4^2)
Simplifying further:
= log(1/8) + log(16)
Using the property that adding logarithms is the same as multiplying the numbers inside the logarithm:
= log((1/8) * 16)
= log(2/8)
= log(1/4)
Therefore, the expression log x – log (8x) + 2 log 4 can be simplified as log(1/4).