Suppose that u=log(2) and v=log(5). Find possible formulas for the following expressions in terms of u and/or v. Your answers should NOT involve any log's.
a) log(0.4)=
b) log(0.08)=
c) log(2500)=
a) u - v
b) u - 2 v
c) 2 u + 4 v
To find the formulas for the given expressions in terms of u and/or v without any logarithms, we can use the properties of logarithms.
a) log(0.4):
We can rewrite 0.4 as 4/10 or 2/5. Using the property log(a/b) = log(a) - log(b), we have:
log(0.4) = log(2/5) = log(2) - log(5) = u - v
Therefore, a possible formula for log(0.4) in terms of u and v is u - v.
b) log(0.08):
Similarly, we can rewrite 0.08 as 8/100 or 2/25. Using the same property as above:
log(0.08) = log(2/25) = log(2) - log(25) = u - 2v
Therefore, a possible formula for log(0.08) in terms of u and v is u - 2v.
c) log(2500):
Since 2500 is a perfect square of 50, we can rewrite it as 50^2. Using the property log(a^b) = b * log(a), we have:
log(2500) = log(50^2) = 2 * log(50)
Now, we can use the property log(a * b) = log(a) + log(b) to further simplify:
log(2500) = 2 * log(50) = 2 * (log(5) + log(10)) = 2 * (v + u)
Therefore, a possible formula for log(2500) in terms of u and v is 2 * (v + u).
To find the formulas for the given expressions without involving logarithms, we need to make use of the properties of logarithms. Specifically, we can use the base conversion formula:
log_b(x) = log_a(x) / log_a(b), where log_a represents the logarithm with base a.
Let's apply this formula to each part:
a) To find log(0.4), we can use the base conversion formula with base 10:
log(0.4) = log(4/10) = log(4) - log(10)
Now, we know that log(10) = 1, so the formula becomes:
log(0.4) = log(4) - log(10) = log(4) - 1
Next, we express log(4) in terms of u and/or v. We can rewrite 4 as 2^2:
log(4) = log(2^2) = 2log(2) = 2u
Therefore, the formula for log(0.4) in terms of u is:
log(0.4) = 2u - 1
b) Similarly, for log(0.08), using the base conversion formula:
log(0.08) = log(8/100) = log(8) - log(100)
Since log(100) = 2 (as 100 = 10^2), the formula becomes:
log(0.08) = log(8) - log(100) = log(8) - 2
Now, express log(8) in terms of u and/or v. We can rewrite 8 as 2^3:
log(8) = log(2^3) = 3log(2) = 3u
Therefore, the formula for log(0.08) in terms of u is:
log(0.08) = 3u - 2
c) For log(2500), we can simplify it first:
log(2500) = log(25 * 100) = log(25) + log(100)
Since log(25) = 2 (as 25 = 5^2), and log(100) = 2 (as 100 = 10^2), the formula becomes:
log(2500) = log(25) + log(100) = 2 + 2
Therefore, the formula for log(2500) is simply:
log(2500) = 4
In summary:
a) log(0.4) = 2u - 1
b) log(0.08) = 3u - 2
c) log(2500) = 4