Simplify the following expressions. Your answers must be exact and in simplest form.
(a) log8(8^−5x−10)=
(b) 12^log12(−5+7q)=
(c) log4(64^k)=
(d) 3^3log3(10−10log3^3)=
I'll do one. You just need to remember that
log(ab) = log(a) + log(b)
log(a^n) n log(a)
So, (c) can be done as follows:
log4(64^k)
= log4(4^3^k)
= log4(4^(3k))
= 3k
since log_b(b) = 1 for any b
I think you have garbled (d).
Try using some parentheses to clarify, or try typing it in at wolframalpha.com to see how it interprets it.
It will recognize log_3(x) as log3x
ok thx!
(a) To simplify the expression log8(8^−5x−10), we can use the logarithm property that states log(base b) (b^x) = x. In this case, the base is 8 and the exponent is -5x-10, which means we can simplify the expression to -5x-10.
(b) To simplify the expression 12^log12(−5+7q), we can again use the logarithm property mentioned earlier: log(base b) (b^x) = x. In this case, the base is 12 and the exponent is -5+7q. So, the expression simplifies to -5+7q.
(c) To simplify the expression log4(64^k), we can rewrite 64 as 4^3 because 4^3 = 64. Therefore, we have log4((4^3)^k), which can be simplified to log4(4^(3k)). Next, we apply another logarithm property, namely log(base b) (b^x) = x, which states that if the base of the logarithm and the base of the exponentiation are the same, then the logarithm can be removed. So, the expression simplifies to 3k.
(d) To simplify the expression 3^3log3(10−10log3^3), we need to work with each term separately. First, we simplify the expression inside the logarithm: 10−10log3^3 = 10−10(3) = 10−30 = -20.
Now, let's substitute this value back into the original expression: 3^3log3(-20). We can further simplify this by using the logarithm property log(base b) (b^x) = x. In this case, the base is 3 and the exponent is -20, which means that the expression simplifies to -20.
So, the simplified expression is -20.