Question

1)1,3^3x-2=3 (2) 16^x-4^x+1-14=0 решить уравнения

Answer 1

1) (13/10)2*x-2=3, 169/100x-2=3, 100*169/100x-100*2=100*3, 169x-200=300,  169x=300+200,  169x=500,   x=500/169.

3)265x-16+1-14=0,  240X=13,  X=13/240

Answer 2

Let's solve each equation step by step:

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

To solve this equation, we'll isolate the exponential term and then use logarithms to find the value of $x$.

Step 1: Divide both sides by 1: $3^{3x-2} = 3$

Step 2: Take the logarithm of both sides (any base can be used, but let's use the natural logarithm, $\ln$): $\ln(3^{3x-2}) = \ln(3)$

Step 3: Use the logarithmic property $\ln(a^b) = b \cdot \ln(a)$: $(3x-2) \cdot \ln(3) = \ln(3)$

Step 4: Divide both sides by $\ln(3)$: $3x-2 = \frac{\ln(3)}{\ln(3)} = 1$

Step 5: Add 2 to both sides to solve for $x$: $3x = 1 + 2 = 3$

Step 6: Finally, divide both sides by 3: $x = \frac{3}{3} = 1$

So, the solution for the first equation is $x = 1$.

2. $16^x - 4^x + 1 - 14 = 0$

Step 1: Combine like terms: $(16^x - 4^x) + (1 - 14) = 0$

Step 2: Simplify further: $(2^4)^x - (2^2)^x - 13 = 0$

Step 3: Use the property $a^{bc} = (a^b)^c$ to simplify: $(2^{4x}) - (2^{2x}) - 13 = 0$

Now, we have a quadratic-like equation in terms of $2^x$. Let's make a substitution to simplify it further:

Let $u = 2^x$. Then the equation becomes:

$u^2 - u - 13 = 0$

This is a quadratic equation. To solve for $u$, we can use the quadratic formula:

$u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 1$, $b = -1$, and $c = -13$.

Step 4: Calculate the discriminant: $D = b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot (-13) = 1 + 52 = 53$

Step 5: Find the square root of the discriminant: $\sqrt{D} = \sqrt{53}$

Step 6: Apply the quadratic formula: $u = \frac{-(-1) \pm \sqrt{53}}{2 \cdot 1}$ $u = \frac{1 \pm \sqrt{53}}{2}$

So, we have two possible values for $u$:

$u_1 = \frac{1 + \sqrt{53}}{2}$ $u_2 = \frac{1 - \sqrt{53}}{2}$

Step 7: Recall that $u = 2^x$. Now we can solve for $x$ in each case:

For $u_1$: $2^x=\frac{1+\sqrt{53}}{2}$ 0 0