Express the following as a logarithm and simplify.
1. log(3x^2 + 11x - 20) - log(3x - 4)
2. 1/6 log3 27 + log3 18 - log3 2
3. log5 cube root of 625
4. 2loga x^3 + 3loga y + 4loga z - 3logz w
1. Using the quotient rule of logarithms, the expression "log(3x^2 + 11x - 20) - log(3x - 4)" can be simplified as follows:
log((3x^2 + 11x - 20)/(3x - 4))
2. To simplify the expression "1/6 log3 27 + log3 18 - log3 2", we can use the rules of logarithms:
1/6 log3 27 + log3 18 - log3 2
1/6 (log3 3^3) + log3 18 - log3 2
1/6 (3) + log3 18 - log3 2
1/2 + log3 18 - log3 2
3. The expression "log5 cube root of 625" can be written as:
log5 ∛(625)
4. The expression "2loga x^3 + 3loga y + 4loga z - 3logz w" can be simplified using the rules of logarithms:
2loga x^3 + 3loga y + 4loga z - 3logz w
loga (x^3)^2 + loga (y^3) + loga (z^4) - loga (z^3w)
This simplification takes advantage of the power rule of logarithms.
Sure! Let's simplify each of the expressions step by step.
1. To express log(3x^2 + 11x - 20) - log(3x - 4) as a single logarithm, we can use the quotient rule of logarithms. According to the quotient rule, log(a) - log(b) is equal to log(a / b).
Therefore, log(3x^2 + 11x - 20) - log(3x - 4) is equal to log((3x^2 + 11x - 20) / (3x - 4)).
2. To simplify 1/6 log3 27 + log3 18 - log3 2, we can apply the properties of logarithms.
First, let's rewrite log3 27 as log3 (3^3) since 27 is equal to 3^3. Using the property logb (a^c) = c logb (a), we get 1/6 log3 (3^3) = 1/6 * 3 = 1/2.
Next, log3 18 can be simplified as log3 (2 * 9), and using the property logb (a * c) = logb(a) + logb(c), we can rewrite it as log3 2 + log3 9.
Finally, combining these results, we have 1/2 + log3 2 - log3 2. The log3 2 terms cancel out, leaving us with just 1/2.
Therefore, 1/6 log3 27 + log3 18 - log3 2 simplifies to 1/2.
3. To express log5 cube root of 625, we can use the property logb (a^(1/n)) = (1/n) logb (a).
The cube root of 625 is 5, since 5 * 5 * 5 = 125, and 125 * 5 = 625. Therefore, the expression can be simplified to (1/3) log5 625.
Since 5^3 is equal to 125, we can rewrite it as (1/3) log5 (5^3) = (1/3) * 3 = 1.
So, log5 cube root of 625 simplifies to 1.
4. For 2loga x^3 + 3loga y + 4loga z - 3logz w, we can again use the properties of logarithms to simplify.
Using the power rule of logarithms, logb (a^c) = c logb (a), we can rewrite 2loga x^3 as loga (x^3)^2 = loga (x^6).
Similarly, 3loga y can be written as loga (y^3) and 4loga z can be written as loga (z^4).
Lastly, using the property logb (a / c) = logb (a) - logb (c), we can rewrite -3logz w as -logz (w^3).
Combining all these terms, we have loga (x^6) + loga (y^3) + loga (z^4) - logz (w^3).
Using the product rule of logarithms which states that logb (a) + logb (c) = logb (a * c), we can simplify further:
loga (x^6) + loga (y^3) + loga (z^4) - logz (w^3) = loga (x^6 * y^3 * z^4) - logz (w^3).
Therefore, 2loga x^3 + 3loga y + 4loga z - 3logz w simplifies to loga (x^6 * y^3 * z^4) - logz (w^3).