Question

If $a$, $b$, and $c$ are positive real numbers such that $a > 10 \ge b \ge c$ and$\frac{\log_8(a+b)}{\log_2 c} + \frac{\log_{27}(a-b)}{\log_3 c} = \frac{2}{3}$then the greatest possible integer value of $a$ is

Answer 1

We can simplify the given equation using the change of base formula:

$\frac{\log(a+b)}{\log 9} \cdot \log 2 + \frac{\log(a-b)}{\log 27} \cdot \log 3 = \frac{2}{3}$

$\frac{\log(a+b)}{2\log 3} \cdot \log 2 + \frac{\log(a-b)}{3\log 3} \cdot \log 3 = \frac{2}{3}$

$\frac{\log(a+b)}{2} + \frac{\log(a-b)}{3} = \frac{2}{3}$

Multiplying both sides by 6, we get:

$3\log(a+b) + 2\log(a-b) = 12$

Using the logarithmic property $\log a^n = n \log a$, we can rewrite this as:

$\log((a+b)^3(a-b)^2) = 12$

Taking the antilogarithm of both sides, we get:

$(a+b)^3(a-b)^2 = 10^{12}$

Since $a > 10 \ge b \ge c$, we can see that $(a+b)^3$ and $(a-b)^2$ are both positive integers.

To maximize $a$, we need to maximize both $(a+b)$ and $(a-b)$.

By trial and error or using a calculator, we can find that when $a = 17$ and $b = 10$, the equation $(a+b)^3(a-b)^2 = 10^{12}$ is satisfied.

Therefore, the greatest possible integer value of $a$ is 17.

Answer 2

We can simplify the given equation using the change of base formula:

$\frac{\log(a+b)}{\log 9} \cdot \log 2 + \frac{\log(a-b)}{\log 27} \cdot \log 3 = \frac{2}{3}$

$\frac{\log(a+b)}{2\log 3} \cdot \log 2 + \frac{\log(a-b)}{3\log 3} \cdot \log 3 = \frac{2}{3}$

$\frac{\log(a+b)}{2} + \frac{\log(a-b)}{3} = \frac{2}{3}$

Multiplying both sides by 6, we get:

$3\log(a+b) + 2\log(a-b) = 12$

Using the logarithmic property $\log a^n = n \log a$, we can rewrite this as:

$\log((a+b)^3(a-b)^2) = 12$

Taking the antilogarithm of both sides, we get:

$(a+b)^3(a-b)^2 = 10^{12}$

Since $a > 10 \ge b \ge c$, we can see that $(a+b)^3$ and $(a-b)^2$ are both positive integers.

To maximize $a$, we need to maximize both $(a+b)$ and $(a-b)$.

By trial and error or using a calculator, we can find that when $a = 17$ and $b = 10$, the equation $(a+b)^3(a-b)^2 = 10^{12}$ is satisfied.

Therefore, the greatest possible integer value of $a$ is 17.