Input:
**User**:
Assume $0 < r < 3$.  Below are five equations for $x$.  Which equation has the largest solution $x$?

$\textbf{(A)}\ 3(1 + r)^x = 7\qquad \textbf{(B)}\ 3(1 + r/10)^x = 7\qquad \textbf{(C)}\ 3(1 + 2r)^x = 7$ $\textbf{(D)}\ 3(1 + \sqrt {r})^x = 7\qquad \textbf{(E)}\ 3(1 + 1/r)^x = 7$
Please reason step by step, and put your final answer within \boxed{}.

<think>
Okay, so I need to figure out which of these equations has the largest solution for x, given that 0 < r < 3. Let me start by understanding what each equation is saying. All of them are of the form 3 times some base raised to the power of x equals 7. So, if I can solve each equation for x, I can compare the solutions.

First, let me recall that for an equation like A * B^x = C, solving for x would involve taking the logarithm of both sides. Let me write that down. If I have 3*(something)^x = 7, then dividing both sides by 3 gives (something)^x = 7/3. Then taking the natural logarithm (or log base 10, doesn't matter as long as I'm consistent) of both sides gives x * ln(something) = ln(7/3), so x = ln(7/3)/ln(something). Therefore, the value of x depends inversely on the logarithm of the base. So, if the base is larger, then ln(base) is larger, so x would be smaller. Conversely, if the base is smaller, ln(base) is smaller, so x would be larger. Therefore, the equation with the smallest base will have the largest x. Wait, is that correct?

Wait, let me check. Suppose I have two equations: A * B^x = C and A * D^x = C. Then solving for x gives x = ln(C/A)/ln(B) and x = ln(C/A)/ln(D). So, if B < D, then ln(B) < ln(D), so x would be larger for the equation with the smaller base. Therefore, yes, the equation with the smallest base will have the largest x. Therefore, to find which equation has the largest x, I need to find which equation has the smallest base. So, the base is (1 + r), (1 + r/10), (1 + 2r), (1 + sqrt(r)), and (1 + 1/r). So, I need to compare these bases for 0 < r < 3 and find which one is the smallest.

But wait, hold on. Let me make sure. Let me think again. Since x is proportional to 1/ln(base), so if the base is smaller, ln(base) is smaller, so 1/ln(base) is larger. Therefore, x is larger. Therefore, the equation with the smallest base will have the largest x. Therefore, the problem reduces to finding which of these bases is the smallest for 0 < r < 3.

So, let me analyze each base:

A: 1 + r

B: 1 + r/10

C: 1 + 2r

D: 1 + sqrt(r)

E: 1 + 1/r

So, given that 0 < r < 3, let's analyze each base.

First, note that r is between 0 and 3. Let me consider each base:

For base A: 1 + r. Since r is between 0 and 3, this base is between 1 and 4.

For base B: 1 + r/10. Since r is between 0 and 3, r/10 is between 0 and 0.3, so base B is between 1 and 1.3.

For base C: 1 + 2r. Since r is between 0 and 3, 2r is between 0 and 6, so base C is between 1 and 7.

For base D: 1 + sqrt(r). Since r is between 0 and 3, sqrt(r) is between 0 and sqrt(3) ≈ 1.732. Therefore, base D is between 1 and approximately 2.732.

For base E: 1 + 1/r. Since r is between 0 and 3, 1/r is between 1/3 ≈ 0.333 and approaching infinity as r approaches 0. However, since r is greater than 0, 1/r is positive. But since r is less than 3, 1/r is greater than 1/3. Therefore, base E is between 1 + 1/3 ≈ 1.333 and approaching infinity as r approaches 0.

So, now, comparing all these bases:

Base B is between 1 and 1.3.

Base E is between approximately 1.333 and infinity.

Base A is between 1 and 4.

Base D is between 1 and ~2.732.

Base C is between 1 and 7.

So, the smallest base is base B? Wait, but base B is between 1 and 1.3, while base E is between 1.333 and infinity. So, base B is smaller than base E. However, base B is between 1 and 1.3, but base D is between 1 and ~2.732. So, base B is smaller than base D. Similarly, base A is between 1 and 4, so base B is smaller than base A. So, base B is the smallest? Wait, but hold on. Wait, base B is 1 + r/10. Since r is between 0 and 3, then base B is between 1 and 1.3. However, base E is 1 + 1/r. When r approaches 0, 1/r approaches infinity, so base E can be very large. But when r approaches 3, base E approaches 1 + 1/3 ≈ 1.333. So, for r in (0, 3), base E is between 1.333 and infinity. Therefore, base B is between 1 and 1.3, which is less than base E's minimum value of ~1.333. Therefore, base B is smaller than base E. Similarly, base B is smaller than base A, D, C. Therefore, base B is the smallest base. Therefore, according to my previous reasoning, equation B would have the largest x. But wait, let me check again.

Wait, but hold on. Wait, if base B is smaller than base E, then equation B would have a larger x than equation E. But let me check if there is any case where base B is not the smallest. For example, when r is very small, say approaching 0. Then base B approaches 1 + 0 = 1, while base E approaches 1 + infinity = infinity. So, base B is smaller. When r approaches 3, base B approaches 1 + 3/10 = 1.3, and base E approaches 1 + 1/3 ≈ 1.333. So, base B is still smaller. Therefore, for all r in (0, 3), base B is the smallest. Therefore, equation B would have the largest x. But wait, let me check the answer options. Wait, the answer options are A, B, C, D, E. But according to this, equation B would have the largest x. However, let me check again. Wait, maybe I made a mistake here.

Wait, but let me think again. Suppose that the base is smaller, so the exponent x is larger. Therefore, equation B has the smallest base, so x is largest. Therefore, answer is B? But let me check with an example. Let me take a specific value of r, say r = 1. Then compute the bases:

For r = 1:

A: 1 + 1 = 2

B: 1 + 1/10 = 1.1

C: 1 + 2*1 = 3

D: 1 + sqrt(1) = 2

E: 1 + 1/1 = 2

So, bases are 2, 1.1, 3, 2, 2. So, base B is the smallest. Therefore, equation B would have the largest x. Let me compute x for each equation with r = 1.

Original equations:

A: 3*(2)^x = 7 => x = ln(7/3)/ln(2) ≈ ln(2.333)/0.693 ≈ 0.847/0.693 ≈ 1.222

B: 3*(1.1)^x = 7 => x = ln(7/3)/ln(1.1) ≈ 0.847 / 0.0953 ≈ 8.89

C: 3*(3)^x = 7 => x = ln(7/3)/ln(3) ≈ 0.847 / 1.0986 ≈ 0.771

D: 3*(2)^x = 7 => same as A, so x ≈ 1.222

E: 3*(2)^x = 7 => same as A and D, so x ≈ 1.222

So, with r = 1, equation B has the largest x. So, that seems to confirm my previous conclusion. But let me check another value of r. Let me take r approaching 0. Let's say r = 0.1.

For r = 0.1:

A: 1 + 0.1 = 1.1

B: 1 + 0.1/10 = 1 + 0.01 = 1.01

C: 1 + 2*0.1 = 1.2

D: 1 + sqrt(0.1) ≈ 1 + 0.316 ≈ 1.316

E: 1 + 1/0.1 = 1 + 10 = 11

So, bases are 1.1, 1.01, 1.2, 1.316, 11. So, base B is the smallest. Therefore, equation B would have the largest x. Let me compute x for each equation with r = 0.1.

Original equations:

A: 3*(1.1)^x = 7 => x = ln(7/3)/ln(1.1) ≈ 0.847 / 0.0953 ≈ 8.89

B: 3*(1.01)^x = 7 => x = ln(7/3)/ln(1.01) ≈ 0.847 / 0.00995 ≈ 85.1

C: 3*(1.2)^x = 7 => x ≈ 0.847 / 0.1823 ≈ 4.646

D: 3*(1.316)^x = 7 => x ≈ 0.847 / 0.275 ≈ 3.08

E: 3*(11)^x = 7 => x = ln(7/3)/ln(11) ≈ 0.847 / 2.398 ≈ 0.353

So, here, equation B has the largest x. So, again, equation B is the largest. Let me check another value, say r approaching 3. Let me take r = 2.9.

For r = 2.9:

A: 1 + 2.9 = 3.9

B: 1 + 2.9/10 = 1 + 0.29 = 1.29

C: 1 + 2*2.9 = 1 + 5.8 = 6.8

D: 1 + sqrt(2.9) ≈ 1 + 1.702 ≈ 2.702

E: 1 + 1/2.9 ≈ 1 + 0.3448 ≈ 1.3448

So, bases are 3.9, 1.29, 6.8, 2.702, 1.3448. So, base B is 1.29, which is smaller than base E's 1.3448. Therefore, equation B still has the smallest base. Let me compute x for each equation with r = 2.9.

A: 3*(3.9)^x = 7 => x = ln(7/3)/ln(3.9) ≈ 0.847 / 1.361 ≈ 0.622

B: 3*(1.29)^x = 7 => x ≈ 0.847 / ln(1.29) ≈ 0.847 / 0.255 ≈ 3.32

C: 3*(6.8)^x = 7 => x ≈ 0.847 / ln(6.8) ≈ 0.847 / 1.916 ≈ 0.442

D: 3*(2.702)^x = 7 => x ≈ 0.847 / ln(2.702) ≈ 0.847 / 0.995 ≈ 0.851

E: 3*(1.3448)^x = 7 => x ≈ 0.847 / ln(1.3448) ≈ 0.847 / 0.298 ≈ 2.84

So, here, equation B has x ≈ 3.32, which is the largest. So, again, equation B is the largest. Therefore, seems like equation B is the answer. But wait, let me check if there is any case where another equation could have a smaller base. For example, equation D: 1 + sqrt(r). Let me see if there is a value of r where 1 + sqrt(r) is smaller than 1 + r/10. Let me set 1 + sqrt(r) < 1 + r/10 => sqrt(r) < r/10 => sqrt(r) - r/10 < 0. Let me solve sqrt(r) < r/10. Let me square both sides (but need to be careful with inequalities). However, since sqrt(r) is positive, and r/10 is positive. Let me consider r > 0. Let me set sqrt(r) < r/10. Multiply both sides by 10: 10 sqrt(r) < r. Let me write this as r - 10 sqrt(r) > 0. Let me set t = sqrt(r), so t > 0. Then the inequality becomes t^2 - 10t > 0 => t(t - 10) > 0. Since t > 0, this inequality holds when t - 10 > 0 => t > 10 => sqrt(r) > 10 => r > 100. But our r is less than 3, so this inequality never holds. Therefore, sqrt(r) >= r/10 for all 0 < r <= 3. Therefore, 1 + sqrt(r) >= 1 + r/10. Therefore, base D is always greater or equal to base B. Therefore, base B is smaller than base D. Therefore, equation B is still the smallest base. Similarly, equation E: 1 + 1/r. Let me check if 1 + r/10 is smaller than 1 + 1/r. Let me set 1 + r/10 < 1 + 1/r => r/10 < 1/r => r^2 < 10 => r < sqrt(10) ≈ 3.16. Since our r is less than 3, which is less than sqrt(10), so this inequality holds. Therefore, for all 0 < r < 3, 1 + r/10 < 1 + 1/r. Therefore, base B is smaller than base E. Therefore, equation B has the smallest base. Therefore, equation B has the largest x. Therefore, answer is B? But wait, let me check equation E again. For example, when r approaches 0, base E approaches infinity, so equation E's x approaches 0. When r approaches 3, base E approaches 1 + 1/3 ≈ 1.333, which is larger than base B's 1.3 when r approaches 3. Therefore, equation B is always smaller. Therefore, equation B is the answer. But wait, the answer options are given as A, B, C, D, E. But let me check again. Wait, the problem says "which equation has the largest solution x". So, according to my analysis, equation B. But let me check if there is any mistake here. Wait, but let me think again. Suppose that for some r, maybe equation D has a smaller base? Wait, earlier I showed that sqrt(r) >= r/10 for all 0 < r <= 3. Therefore, 1 + sqrt(r) >= 1 + r/10. Therefore, base D is larger than base B. Therefore, equation B is the smallest base. Therefore, equation B has the largest x. Therefore, answer is B. But wait, let me check the answer options again. Wait, the answer options are:

(A) 3(1 + r)^x = 7

(B) 3(1 + r/10)^x = 7

(C) 3(1 + 2r)^x = 7

(D) 3(1 + sqrt(r))^x = 7

(E) 3(1 + 1/r)^x = 7

So, according to my analysis, answer is B. However, I need to check if there is any mistake in my reasoning. Let me think again. Suppose that for some r, maybe equation E has a smaller base? For example, if r is very large, but r is less than 3. Wait, for r approaching 3, base E approaches 1 + 1/3 ≈ 1.333, while base B approaches 1 + 3/10 = 1.3. Therefore, base B is smaller. Therefore, equation B is still smaller. Therefore, equation B is the smallest base. Therefore, answer is B. However, let me check if there is any other equation with a smaller base. For example, equation D: 1 + sqrt(r). For r approaching 0, sqrt(r) approaches 0, so base D approaches 1. But base B approaches 1 + 0 = 1. So, for r approaching 0, both bases approach 1. However, for r approaching 0, base B is 1 + r/10, which is slightly larger than 1, and base D is 1 + sqrt(r), which is also slightly larger than 1. But which one is smaller? Let me take r approaching 0. Let me take r = 0.01. Then base B is 1 + 0.01/10 = 1.001. Base D is 1 + sqrt(0.01) = 1 + 0.1 = 1.1. Therefore, base B is smaller. Therefore, even for small r, base B is smaller. Therefore, equation B is the smallest base. Therefore, answer is B. However, let me check if equation E can ever be smaller. For example, if r is very small, say r approaches 0, then base E approaches infinity, so no. If r approaches 3, base E approaches 1.333, which is larger than base B's 1.3. Therefore, equation B is always smaller. Therefore, answer is B. However, I need to check if the problem says "the largest solution x". Wait, but let me think again. Suppose that for some r, maybe equation D has a smaller base? Wait, earlier I showed that sqrt(r) >= r/10 for all 0 < r <= 3. Therefore, 1 + sqrt(r) >= 1 + r/10. Therefore, base D is larger than base B. Therefore, equation B is the smallest base. Therefore, answer is B. Therefore, the answer should be B. However, I need to check if there is any mistake here. Wait, but let me check with another example. Let me take r = 0. Let me take r approaching 0. Let me take r = 0.0001. Then base B is 1 + 0.0001/10 = 1.00001. Base D is 1 + sqrt(0.0001) = 1 + 0.01 = 1.01. Therefore, base B is smaller. Therefore, equation B is still the smallest. Therefore, answer is B. Therefore, I think the answer is B. However, let me check the answer options again. Wait, but the original problem says "Assume 0 < r < 3". So, r is strictly between 0 and 3. Therefore, there is no problem with r = 0 or r = 3. Therefore, my analysis holds. Therefore, equation B has the smallest base, so the largest x. Therefore, answer is B. However, I need to check if the answer is different. Wait, but let me check the answer options again. Wait, the answer options are given as A, B, C, D, E. But maybe I made a mistake in the initial reasoning. Let me think again. Suppose that the base is smaller, so the exponent x is larger. Therefore, the equation with the smallest base will have the largest x. Therefore, equation B. However, let me check if there is any other equation with a smaller base. For example, equation E: 1 + 1/r. For r approaching 0, base E approaches infinity. For r approaching 3, base E approaches 1 + 1/3 ≈ 1.333. So, base E is always larger than base B. Therefore, equation B is the smallest. Therefore, answer is B. Therefore, I think the answer is B. However, I need to check if the answer is different. Wait, but let me check with another approach. Let me take the general case. Let me suppose that for all 0 < r < 3, which base is the smallest. Let me compare each pair:

Compare base B (1 + r/10) with base A (1 + r). Since r/10 < r for 0 < r < 3, so base B < base A.

Compare base B with base C (1 + 2r). Since r/10 < 2r for 0 < r < 3? Let me check. For r > 0, r/10 < 2r is equivalent to 1/10 < 2, which is true. Therefore, base B < base C.

Compare base B with base D (1 + sqrt(r)). As shown earlier, sqrt(r) >= r/10 for 0 < r <= 3. Therefore, base D >= base B.

Compare base B with base E (1 + 1/r). As shown earlier, for 0 < r < 3, 1 + r/10 < 1 + 1/r. Therefore, base B < base E.

Therefore, base B is the smallest. Therefore, equation B has the largest x. Therefore, answer is B. Therefore, the answer is B. However, I need to check if the answer is different. Wait, but let me check the original problem again. Wait, the problem says "which equation has the largest solution x". So, according to my analysis, answer is B. However, I need to check if there is any mistake in my reasoning. Wait, but let me think again. Suppose that the base is smaller, so the exponent x is larger. Therefore, equation with the smallest base has the largest x. Therefore, equation B. Therefore, answer is B. Therefore, I think the answer is B. However, I need to check if the answer is different. Wait, but maybe I made a mistake in the initial step. Let me check with another example. Let me take r = 0.5. Then:

Base A: 1 + 0.5 = 1.5

Base B: 1 + 0.5/10 = 1.05

Base C: 1 + 2*0.5 = 2

Base D: 1 + sqrt(0.5) ≈ 1 + 0.707 ≈ 1.707

Base E: 1 + 1/0.5 = 1 + 2 = 3

So, bases are 1.5, 1.05, 2, 1.707, 3. Therefore, base B is the smallest. Therefore, equation B has the largest x. Let me compute x for each equation with r = 0.5.

Original equations:

A: 3*(1.5)^x = 7 => x = ln(7/3)/ln(1.5) ≈ 0.847 / 0.405 ≈ 2.09

B: 3*(1.05)^x = 7 => x ≈ 0.847 / ln(1.05) ≈ 0.847 / 0.04879 ≈ 17.37

C: 3*(2)^x = 7 => x ≈ 0.847 / 0.693 ≈ 1.222

D: 3*(1.707)^x = 7 => x ≈ 0.847 / ln(1.707) ≈ 0.847 / 0.536 ≈ 1.58

E: 3*(3)^x = 7 => x ≈ 0.847 / 1.0986 ≈ 0.771

So, equation B has the largest x. Therefore, again, answer is B. Therefore, I think my conclusion is correct. Therefore, the answer is B. However, I need to check if the answer is different. Wait, but let me check the answer options again. Wait, the answer options are labeled A to E. So, answer is B. However, I need to check if the original problem had any typos or if I misread. Wait, the problem says "Assume 0 < r < 3". So, r is between 0 and 3. Therefore, my analysis holds. Therefore, answer is B. Therefore, I think the answer is B. However, I need to check once more. Wait, but let me think about the function x = ln(7/3)/ln(base). Since ln(base) is positive because base > 1 (since 0 < r < 3, so for example, base B is 1 + r/10. Since r > 0, base B > 1. Similarly, all other bases are greater than 1? Let me check:

For base A: 1 + r. Since r > 0, base A > 1.

For base B: 1 + r/10. Since r > 0, base B > 1.

For base C: 1 + 2r. Since r > 0, base C > 1.

For base D: 1 + sqrt(r). Since r > 0, sqrt(r) > 0, so base D > 1.

For base E: 1 + 1/r. Since r > 0, 1/r > 0, so base E > 1.

Therefore, all bases are greater than 1, so ln(base) is positive, so x is positive. Therefore, the previous reasoning holds. Therefore, equation B has the smallest base, so the largest x. Therefore, answer is B. Therefore, I think the answer is B. However, I need to check if the answer is different. Wait, but maybe I made a mistake in the initial step. Let me think again. Suppose that the equation is 3*(base)^x = 7. Then solving for x gives x = log_{base}(7/3). Therefore, x = log_{base}(7/3). Therefore, the larger the base, the smaller the logarithm. Therefore, the smaller the base, the larger the logarithm. Therefore, equation with the smallest base has the largest x. Therefore, answer is B. Therefore, I think the answer is B. Therefore, I think the answer is B. However, I need to check if the answer is different. Wait, but maybe the answer is E? Wait, no. For example, when r approaches 0, base E approaches infinity, so log_{base}(7/3) approaches 0. Therefore, x approaches 0. 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Solution:
Intuitively, $x$ will be largest for the option for which the value in the parentheses is smallest.

Formally, first note that each of the values in parentheses is larger than $1$. Now, each of the options is of the form $3f(r)^x = 7$. This can be rewritten as $x\log f(r) = \log\frac 73$. As $f(r)>1$, we have $\log f(r)>0$. Thus $x$ is the largest for the option for which $\log f(r)$ is smallest. As $\log f(r)$ is an increasing function, this is the option for which $f(r)$ is smallest.

We now get the following easier problem: Given that $0<r<3$, find the smallest value in the set $\{ 1+r, 1+r/10, 1+2r, 1+\sqrt r, 1+1/r\}$.

Clearly, $1+r/10$ is smaller than the first and the third option.

We have $r^2 < 10$, so dividing both sides by $10r$, we get $r/10 < 1/r$.

Finally, $r/100 < 1$, therefore $r^2/100 < r$. Since both sides are positive, we can take the square root and get $r/10 < \sqrt r$.

Thus the answer is $\boxed{\text{(B)}} 3(1 + r/10)^x = 7$.