Simplify logx(x/4)+3logx(2/5)-log21(2/25)
is that last term log base 21?
whatever - we can start with
log_x(x/4) = log_x(x) - log_x(4) = 1-log_x(4)
So, assuming all logs to the same base (x), we can make it
1+log(1/4 * 8/125 * 25/2) = 1-2log5
I suspect multiple typos
To simplify the expression logx(x/4) + 3logx(2/5) - log21(2/25), we need to use logarithmic properties to simplify each term separately and then combine them.
1. Simplify logx(x/4):
According to the logarithmic property log_a(b/c) = log_a(b) - log_a(c), we can rewrite logx(x/4) as logx(x) - logx(4):
logx(x/4) = logx(x) - logx(4)
Now, logx(x) is equal to 1 since the logarithm base is x and x raised to the power of 1 results in x.
Therefore, logx(x) = 1.
logx(4) represents the power to which x must be raised to obtain the value 4. This can be expressed as:
x^y = 4, where x is the base and y is the exponent. Now we solve for y:
x^y = 4
By using the logarithmic property, log_a(b) = y, we get:
logx(4) = y
Therefore, logx(4) = y = log4(x)
Combining these results, we have:
logx(x/4) = 1 - log4(x)
2. Simplify 3logx(2/5):
Using the logarithmic property log_a(b^c) = c log_a(b), we can rewrite 3logx(2/5) as logx((2/5)^3):
3logx(2/5) = logx((2/5)^3)
Now, (2/5)^3 represents the cube of 2/5. This can be calculated as:
(2/5)^3 = 2^3 / 5^3 = 8/125
Therefore, we have:
3logx(2/5) = logx(8/125)
3. Simplify log21(2/25):
Since the logarithm base is 21, we cannot directly simplify log21(2/25) since 2/25 is not a power of 21.
Combining the simplified terms, we get:
logx(x/4) + 3logx(2/5) - log21(2/25) = 1 - log4(x) + logx(8/125) - log21(2/25)