log3^x=p and log18^x=q show that log6^3=q divide by p-q
log3^x=p ---> xlog3 = p
log18^x=q --> xlog18 = q
we want to show q/(p-q) = log6^3
p-q = xlog 3 - xlog18
= x(log3 - log18)
= x(log (3/18) = x log(1/6) = x(log1 - log6)
= x(0- log6) = -xlog6
the q/(p-q)
= xlog3/(-xlog6)
= - log3/log6 ≠ log6^3
something is not right with your question.
To solve the given problem, let's start by rewriting the logarithmic expressions using the properties of logarithms:
For the first expression, log3^x = p, we can rewrite it as:
x * log3 = p
Similarly, for the second expression, log18^x = q, we can rewrite it as:
x * log18 = q
Now, we need to show that log6^3 = q / (p - q). To do this, we can convert all the logarithms involved to a common base, such as base 3:
Using the change of base formula, we can rewrite log6^3 as:
log6^3 = log(6^3)/log(3)
Since 6^3 = 216, we have:
log6^3 = log(216) / log(3)
Now, let's simplify it further by using the properties of logarithms:
log(216) / log(3) = (log(2^3 * 3^3) / log(3)
Using the properties of logarithms again, we can split this expression as:
(log2^3 + log3^3) / log(3)
And since log3^3 = 3 * log3, we can rewrite it as:
(log2^3 + 3 * log3) / log(3)
Substituting the values we already have for log3^x and log18^x, we get:
(log2^3 + 3 * log3) / log(3) = (p + 3q) / log(3)
Thus, we have shown that log6^3 = (p + 3q) / log(3).
To divide by p - q, we simply multiply both the numerator and the denominator by log(3):
(log2^3 + 3 * log3) / log(3) * log(3) / (p - q) = (p + 3q) / (p - q)
Hence, we have proven that log6^3 = q / (p - q).
To prove that log6^3 = q divided by p-q, we will begin by expressing log3^x = p and log18^x = q in terms of a common base, such as 10.
1. Start with the given equations:
log3^x = p (equation 1)
log18^x = q (equation 2)
2. Rewrite the equations using the change-of-base formula:
log3^x = log(x) / log(3) (equation 1a)
log18^x = log(x) / log(18) (equation 2a)
3. Use logarithmic properties to simplify the expressions:
log18^x = log(3^2 * 2)^x = log(3^2) * log(2^x) = 2 * (log(x) / log(3)) (equation 2b)
4. Since we have expressions in terms of log(x) / log(3), we can now equate equation 1a and equation 2b:
log(x) / log(3) = 2 * (log(x) / log(3))
log(x) / log(3) = 2 * log(x) / log(3)
5. Simplify by canceling out log(x) / log(3) from both sides:
1 = 2
6. The equation 1 = 2 is not true, so the given statements "log3^x = p" and "log18^x = q" cannot be simultaneously true. Therefore, we cannot proceed with dividing by p-q, as the equality does not hold.
In conclusion, the statement log6^3 = q divided by p-q cannot be proved, as the initial equations given do not hold true simultaneously.