Question

3^5х-1 *7^2х-2 = 3^3х+1 решите степенное уравнение

Answer 1

$3^{5x-1}*7^{2x-2}=3^{3x+1}|:3^{3x+1}>0\\\\\frac{3^{5x-1}*7^{2x-2}}{3^{3x+1}}=\frac{3^{3x+1}}{3^{3x+1}}\\\\3^{5x-1-3x-1} *7^{2x-2}=1\\\\3^{2x-2}*7^{2x-2}=1\\\\(3*7)^{2x-2}=1\\\\21^{2x-2}=21^{o}\\\\2x-2=0\\\\2x=2\\\\x=1\\\\Otvet:\boxed{1}$

Answer 2

To solve the exponential equation $3^{5x-1} \cdot 7^{2x-2} = 3^{3x+1}$, we can use the properties of exponents and logarithms. First, we will rewrite the equation in terms of a single base, either 3 or 7, and then solve for $x$.

Let's work with the base 3:

$3^{5x-1} \cdot 7^{2x-2} = 3^{3x+1}$

Now, we'll use the property of exponents that states $a^m \cdot a^n = a^{m+n}$:

$3^{5x-1} \cdot 7^{2x-2} = 3^{3x} \cdot 3^1$

Next, we can use the property of exponents that states $a^1 = a$:

$3^{5x-1} \cdot 7^{2x-2} = 3^{3x} \cdot 3$

Since the bases are equal (both are 3), we can equate the exponents:

$5x - 1 + 2x - 2 = 3x + 1$

Now, solve for $x$:

$7x - 3 = 3x + 1$

Subtract $3x$ from both sides:

$4x - 3 = 1$

Add $3$ to both sides:

$4x = 4$

Finally, divide by $4$ to isolate $x$:

$x = 1$

So, the solution to the exponential equation is $x = 1$.