Question

Log3 (5x-6) < 3+log3(2)

Answer 1

5x-6<5
5x<11
x<2.2
.................

Answer 2

ОДЗ: 5х-6>0
5х>6
Х>1,2

3>1-следовательно знак не меняем
5х-6<54
5х<60
Х<12
Ответ: х принадлежит (1,2;12)

Answer 3

To solve the inequality $\log_3(5x - 6) < 3 + \log_3(2)$, we can use logarithm properties and algebraic manipulation. Here are the steps to solve it:

1. Start with the given inequality: $\log_3(5x - 6) < 3 + \log_3(2)$

2. Subtract $\log_3(2)$ from both sides of the inequality to isolate the logarithmic term on the left side: $\log_3(5x - 6) - \log_3(2) < 3$

3. Use the properties of logarithms to combine the two logarithmic terms on the left side by using the quotient rule, which states that $\log_a(b) - \log_a(c) = \log_a(b/c)$: $\log_3\left(\frac{5x - 6}{2}\right) < 3$

4. Now, we need to remove the logarithm. To do this, we can rewrite the inequality in exponential form: $3^3 > \frac{5x - 6}{2}$

5. Simplify the left side by evaluating $3^3$: $27 > \frac{5x - 6}{2}$

6. Multiply both sides of the inequality by 2 to eliminate the fraction: $54 > 5x - 6$

7. Add 6 to both sides of the inequality to isolate $5x$: $5x < 54 + 6$

8. Simplify the right side: $5x < 60$

9. Finally, divide both sides by 5 to solve for $x$: $x < \frac{60}{5}$

Simplify the right side: $x < 12$

So, the solution to the inequality is $x < 12$.