Logarithm Calculation: Solve Loga(5√(x) * Y^3)

by SLV Team 47 views

Let's break down how to solve this logarithm problem step by step, guys. This involves using the properties of logarithms and some algebraic manipulation. We're given that logaX = 2 and logxY = 3, and our mission is to find the value of loga(5√(x) * Y^3). Grab your thinking caps, and let's dive in!

Understanding the Basics

Before we get started, it’s crucial to understand some fundamental logarithm properties. Logarithms are essentially the inverse of exponential functions. The expression logaX = 2 means that a^2 = X. Similarly, logxY = 3 means that x^3 = Y. These relationships are the building blocks for solving our problem.

Also, remember these key logarithm properties:

  1. Product Rule: loga(MN) = logaM + logaN
  2. Quotient Rule: loga(M/N) = logaM - logaN
  3. Power Rule: loga(M^k) = k * logaM
  4. Change of Base Rule: logbM = logaM / logab

With these rules in mind, we can start dissecting the problem at hand.

Step-by-Step Solution

1. Express the Target Expression

Our target expression is loga(5√(x) * Y^3). We can use the product rule to break this down:

loga(5√(x) * Y^3) = loga5 + loga√(x) + loga(Y^3)

2. Simplify the Terms

Now, let's simplify each term individually.

  • loga5: This term remains as is for now because we don't have direct information about loga5.
  • loga√(x): Since √(x) = x^(1/2), we can rewrite this as loga(x^(1/2)). Using the power rule, this becomes (1/2) * logaX. We know that logaX = 2, so this term simplifies to (1/2) * 2 = 1.
  • loga(Y^3): Using the power rule, this becomes 3 * logaY. We need to find logaY.

3. Find logaY

We know logxY = 3 and logaX = 2. We can use the change of base rule to relate these.

logxY = logaY / logaX

Plugging in the known values:

3 = logaY / 2

Solving for logaY:

logaY = 3 * 2 = 6

4. Substitute Back into the Expression

Now we substitute the values we found back into our original expression:

loga(5√(x) * Y^3) = loga5 + loga√(x) + loga(Y^3) = loga5 + 1 + 3 * logaY

loga(5√(x) * Y^3) = loga5 + 1 + 3 * 6 = loga5 + 1 + 18 = loga5 + 19

5. Address the Constant Term

Oops! It looks like there was an oversight in the original problem statement or alternatives. The expression loga(5√(x) * Y^3) simplifies to loga5 + 19. Since the value of loga5 is not provided and cannot be derived from the given information, we can infer that the question might have intended loga(√(x) * Y^3) instead of loga(5√(x) * Y^3). If that's the case, we proceed as follows:

loga(√(x) * Y^3) = loga√(x) + loga(Y^3) = 1 + 3 * logaY = 1 + 3 * 6 = 1 + 18 = 19

However, considering the provided alternatives, it's likely that the '5' was not intended to be part of the logarithm. Thus, let's re-evaluate based on what we have.

If the expression was meant to be loga(√(x) * Y^3), then:

loga(√(x) * Y^3) = loga(x^(1/2) * Y^3) = loga(x^(1/2)) + loga(Y^3)

= (1/2)loga(x) + 3loga(Y) = (1/2) * 2 + 3 * 6 = 1 + 18 = 19

Given the options, and after a careful review, there seems to be a mistake in the provided options or the question itself. However, let's consider a slightly different approach to match one of the options.

Let’s assume the question is asking for loga(x * Y^3) instead of loga(5√(x) * Y^3).

loga(x * Y^3) = loga(x) + loga(Y^3) = loga(x) + 3loga(Y)

We know loga(x) = 2 and loga(Y) = 6, so:

loga(x * Y^3) = 2 + 3 * 6 = 2 + 18 = 20

Since 20 is not among the options, we look for other possible interpretations.

Re-Examine Possible Correct Scenarios

If the question actually meant loga(x^(1/2) * Y^3) (without the 5), we already found:

loga(x^(1/2) * Y^3) = 19

But let's check if there's a typo and one of the numbers is slightly different.

If the question was intended to be logx(√(a) * Y^3) and we want to express it using the given values:

logx(√(a) * Y^3) = logx(a^(1/2)) + logx(Y^3) = (1/2)logx(a) + 3logx(Y)

Using the change of base: loga(x) = 2 implies logx(a) = 1/2

logx(√(a) * Y^3) = (1/2)(1/2) + 33 = 1/4 + 9 = 9.25 (Not an integer, so unlikely)

After thorough review and considering possible interpretations and likely intended question, let's assume a typo in the constants, and see if any option makes sense with minimal changes. The closest plausible answer by changing a minimal amount is obtained when considering loga(x * Y^2) instead of the cube. In this case:

loga(x * Y^2) = loga(x) + 2loga(Y) = 2 + 26 = 2 + 12 = 14* (Still not among the options).

Conclusion

Given the original expression loga(5√(x) * Y^3) and the provided alternatives, none of the options seem to directly match the result. There is likely an error or missing information in the problem statement. However, under the assumption that the question intended to ask for loga(√(x) * Y^3), the answer would be 19. If the '5' was erroneously included, there may be an unresolvable issue without further assumptions. Please review the question for any possible typos or missing data.

Without additional clarification, it's impossible to definitively select A, B, C, or D. However, the most plausible calculation (assuming a slightly different intended question) leads to a value near 18 or 19, making option D (18) the closest reasonable choice given the limitations. If loga5 was negligible or the '5' was a typo, this might be the intended answer.