simplify the expression
2log(4)9-log(2)3
*(base)
i know the answer is log(2)3 but i can't figure out the work to get that answer
Use the rules for changing the base of a logarithm at
http://oakroadsystems.com/math/loglaws.htm#NewBase
It says that
log(a)x = log(b)x/log(b)a
In your case, use it to show that
log(4)9 = log(2)3^2/log(2)4
= 2log(2)3/log(2)4 = log(2)3 !!
Therefore 2log(4)9 = 2 log(2)3
Subtracting 2log(2)3 from that leaves you with log(2)3
The last line should read:
Subtracting log(2)3 from 2log(2)3 leaves you with log(2)3
To simplify the expression 2log(4)9 - log(2)3, we can use logarithmic properties. Let's break it down step by step:
Step 1: Apply the Power Rule of Logarithms
The Power Rule states that log(base a)(b^c) = c * log(base a)(b). This rule allows us to move the exponent as a coefficient in front of the logarithm.
In this case, we have:
2log(4)9 - log(2)3
Using the Power Rule, we can rewrite each logarithm as:
= log(4)9^2 - log(2)3
Step 2: Simplify the exponents
We can simplify the exponents inside each logarithm:
= log(4)81 - log(2)3
Step 3: Evaluate the logarithms
To evaluate the logarithms, we need to determine the bases. In this case, we have base 4 and base 2.
Let's solve for each logarithm separately:
log(4)81:
To find the value of log(4)81, we need to determine what exponent (power) 4 should be raised to in order to get 81.
4^x = 81
We can rewrite 81 as a power of 4:
4^x = 4^4
Since the bases are the same, the exponents must be equal:
x = 4
Therefore, log(4)81 = 4.
log(2)3:
To find the value of log(2)3, we need to determine what exponent (power) 2 should be raised to in order to get 3.
2^x = 3
Since we can't find a simple integer value for x that satisfies this equation, the expression log(2)3 cannot be simplified further.
Hence, the final simplified expression is:
2log(4)9 - log(2)3 = 2(4) - log(2)3 = 8 - log(2)3 = log(2)3.