CAT 2023 Slot 2 QA Question 7

The question says: For some positive real number x, if log base \ of x plus (log base x of 25 divided by log base x of 0.008) equals 16/3, then we need to find the value of log base 3 of (3x²). The correct answer is given as 7, but I need to work through it.

First, We have the equation as given:

log_{√3}(x) + [log_x(25) / log_x(0.008)] = 16/3

And we need to find log_3(3x²).

Hmm, okay. So, logarithms can be converted from one base to another using the change of base formula. Maybe that will help here.

Let me recall the change of base formula: log_b(a) = log_c(a) / log_c(b). So, if I have a logarithm in base b, I can express it in terms of base c by dividing the log of a by the log of b in base c.

Looking at the first term, log_{√3}(x). Maybe I can express this in terms of log base 3, since the answer we need is in terms of log base 3.

So, log_{√3}(x) can be written as log_3(x) / log_3(√3). Since log_3(√3) is log_3(3^(1/2)) which is 1/2. So, log_{√3}(x) = 2 * log_3(x). That simplifies the first term.

So, now the equation becomes:

2 * log_3(x) + [log_x(25) / log_x(0.008)] = 16/3

Okay, now let's look at the second part: [log_x(25) / log_x(0.008)]. Hmm, that looks a bit complicated. Maybe I can simplify this fraction.

log_x(a) / log_x(b) is equal to log_b(a). Wait, is that right? Let me verify.

Yes, because log_x(a) = 1 / log_a(x), so log_x(a) / log_x(b) = (1 / log_a(x)) / (1 / log_b(x)) ) = log_b(a) / log_a(x) * log_x(a) ??? Wait, maybe I'm confusing something.

Wait, no. Let me think again. If I have log_x(a) divided by log_x(b), that's equal to log_b(a). Because log_x(a) = ln(a)/ln(x) and log_x(b) = ln(b)/ln(x), so when you divide them, the ln(x) cancels out, and you get ln(a)/ln(b) which is log_b(a). Yes, that's correct.

So, [log_x(25) / log_x(0.008)] = log_{0.008}(25)

So, now the equation becomes:

2 * log_3(x) + log_{0.008}(25) = 16/3

Hmm, okay. Now, Computing log_{0.008}(25). That might be a bit tricky because 0.008 is a decimal. Let me express 0.008 as a fraction. 0.008 is 8/1000, which simplifies to 1/125. So, 0.008 = 1/125.

So, log_{0.008}(25) = log_{1/125}(25)

log_{1/a}(b) = - log_a(b). So, log_{1/125}(25) = - log_{125}(25)

So, now we have:

2 * log_3(x) - log_{125}(25) = 16/3

Now, Computing log_{125}(25). 125 is 5³, and 25 is 5². So, log_{5³}(5²) = (2/3) because log_{a^c}(a^d) = d/c.

So, log_{125}(25) = 2/3. Therefore, log_{0.008}(25) = -2/3.

So, substituting back into the equation:

2 * log_3(x) - (2/3) = 16/3

Let me solve for log_3(x). First, add (2/3) to both sides:

2 * log_3(x) = 16/3 + 2/3 = 18/3 = 6

So, 2 * log_3(x) = 6

Divide both sides by 2:

log_3(x) = 3

So, log base 3 of x is 3. That means x = 3³ = 27.

Wait, let me confirm that. If log_3(x) = 3, then x = 3^3 = 27. Yes, that's correct.

Now, we need to find log_3(3x²). Computing that.

First, substitute x = 27 into 3x²:

3x² = 3(27)² = 3729 = 2187

So, log_3(2187). Hmm, . 3^7 = 2187 because 3^5=243, 3^6=729, 3^7=2187. So, log_3(2187) = 7.

Alternatively, we can compute it step by step without calculating 3x² directly.

log_3(3x²) = log_3(3) + log_3(x²) = 1 + 2*log_3(x)

We already know that log_3(x) = 3, so:

1 + 2*3 = 1 + 6 = 7

Yes, that's correct. So, the value is 7.

Let me recap the steps to make sure I didn't make any mistakes:

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