Let a, b, c be the three distinct positive real numbers such that (2a)^(log_e a) = (bc)^(log_e b) an

Q: Why can't we just take the log of both sides and solve for $a, b, c$ directly?

That is a great instinct! However, you have three variables ($a, b, c$) but only two equations. This means you cannot find the individual values of $a, b, c$ directly. Instead, you must find the algebraic relationships between them, which allows you to simplify the final expression $6a + 5bc$.

Q: Wait, why do we reject the case $y - x = 0$?

The problem explicitly states that $a, b, c$ are distinct positive real numbers. If $y - x = 0$, then $\ln b - \ln a = 0$, which implies $\ln b = \ln a$, so $b = a$. This contradicts the condition that the numbers are distinct. Therefore, we must reject this case.

Q: Is there a trick to knowing which case to pick ($y + x = 0$ or $y + z = 0$)?

In competitive exams, you should test the consistency of the system. If you test $y + x = 0$, you will find it leads to a non-constant expression for $6a + 5bc$. The case $y + z = 0$ is the one that leads to a consistent, constant numerical answer, which is what the problem expects.

Question

Let $a, b, c$ be the three distinct positive real numbers such that $(2a)^{\log_e a} = (bc)^{\log_e b}$ and $b^{\log_e 2} = a^{\log_e c}$. Then $6a + 5bc$ is equal to ______.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The given system of equations is:
$$(2a)^{\ln a} = (bc)^{\ln b}$$
$$b^{\ln 2} = a^{\ln c}$$

These equations appear complex, but they are designed to be simplified through logarithmic transformation. Our objective is to isolate the variables and solve for the required expression.

### The Logarithmic Bridge

To bring the exponents down, we apply the natural logarithm ($\ln$) to both sides of the equations. Using the power rule $\ln(m^n) = n \ln m$, the first equation becomes:
$$\ln a \cdot \ln(2a) = \ln b \cdot \ln(bc)$$

Expanding this using the product rule $\ln(mn) = \ln m + \ln n$, we obtain:
$$\ln a (\ln 2 + \ln a) = \ln b (\ln b + \ln c)$$

### The Substitution Strategy

To simplify the notation, we define the following substitutions:
$x = \ln a$, $y = \ln b$, and $z = \ln c$.

The system now transforms into standard algebraic form:
$$x(\ln 2 + x) = y(y + z)$$
$$\ln 2 \cdot y = x \cdot z$$

Since $\ln 2$ is a constant, we isolate it from the second equation:
$$\ln 2 = \frac{xz}{y}$$

### The Algebraic Dance

We substitute the expression for $\ln 2$ back into the first equation:
$$x\left(\frac{xz}{y} + xight) = y(y + z)$$

Distributing $x$ and multiplying the entire equation by $y$ to clear the denominator yields:
$$x^2z + x^2y = y^3 + y^2z$$

Rearranging the terms to one side, we get:
$$y^3 - x^2y + y^2z - x^2z = 0$$

Grouping the terms allows for factorization:
$$y(y^2 - x^2) + z(y^2 - x^2) = 0$$
$$(y^2 - x^2)(y + z) = 0$$

Applying the difference of squares identity, we arrive at:
$$(y - x)(y + x)(y + z) = 0$$

### The Final Reveal

Given that $a$ and $b$ are distinct, we know $x 
eq y$, which implies $(y - x) 
eq 0$. This leaves us with two possibilities: $y + x = 0$ or $y + z = 0$.

Testing the path $y + z = 0$, we find $\ln b + \ln c = 0$, which implies $\ln(bc) = 0$, or $bc = 1$. Substituting $z = -y$ into our expression for $\ln 2$:
$$\ln 2 = \frac{x(-y)}{y} = -x$$

Since $x = \ln a$, we have $\ln 2 = -\ln a = \ln(a^{-1})$, which leads to $a = 1/2$. We now calculate the final value:
$$6a + 5bc = 6\left(\frac{1}{2}ight) + 5(1) = 3 + 5 = 8$$

The final answer is **8**.

Animated Solution for Mathematics - Basic Mathematics: Let $a, b, c$ be the three distinct positive real numbers such that $(2 a)^{l o g_{e} a} = (b c)^{l o g_{e} b}$ and $b^{l o g_{e} 2} = a^{l o g_{e} c}$ . Then $6 a + 5 b c$ is equal to ______.

Enter Numerical Value:

Visualized Solution

Problem Setup

Simplifying Equation 1

Simplifying Equation 2

Variable Substitution

Combining the Equations

Rearranging Terms

Complete Factorization

Analyzing the Cases

Finding the Value of $a$

Finding the Product $b c$

Final Calculation

The Sigma Insight: Properties of Logarithms

Analyzing the Setup

The Logarithmic Bridge

The Substitution Strategy

The Algebraic Dance

The Final Reveal

Similar Questions

Let $a = 32$ and $b = \frac{1}{5 ^{1/6} 6}$ . If $x, y \in R$ are such that $3 x + 2 y = lo g_{a} (18)^{5/4}$ and $2 x - y = lo g_{b} (1080)$ , then $4 x + 5 y$ is equal to ________.

The value of $((lo g_{2} 9)^{2})^{\frac{1}{l o g _{2} ( l o g _{2} 9 )}} \times (7)^{\frac{1}{l o g _{4} 7}}$ is ________.

If $n$ is a natural number such that $n = p_{1}^{α_{1}} \cdot p_{2}^{α_{2}} \cdot p_{3}^{α_{3}} \dots p_{k}^{α_{k}}$ and $p_{1}, p_{2}, \dots p_{k}$ are distinct primes, then show that $ln n \geq k ln 2$ .

The least value of the expression $2 lo g_{10} x - lo g_{x} (0.01)$ , for $x > 1$ , is

The number $lo g_{2} 7$ is