Question

Log1/2(x^2-10x+9)=0 Log7 28-log7 4+log7 49

Answer 1

Повторите свойства логарифма

Answer 2

To solve the equation Log(1/2)(x^2 - 10x + 9) = 0, we need to rewrite it using logarithm properties.

1. Property: log_b(1) = 0 for any positive base b.
2. Property: log_b(x^n) = n * log_b(x) for any positive base b and positive integer n.

Using these properties, we can rewrite the equation as:

log(1/2)(x^2 - 10x + 9) = log(1/2)(1)

Since the base of the logarithm is 1/2, we can convert it to an exponential form:

x^2 - 10x + 9 = (1/2)^1

Simplify the right side:

x^2 - 10x + 9 = 1/2

Now, let's solve the equation. First, get rid of the fraction by multiplying the entire equation by 2:

2(x^2 - 10x + 9) = 1

Expand the left side:

2x^2 - 20x + 18 = 1

Move all the terms to one side of the equation:

2x^2 - 20x + 17 = 0

Now, we have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = -20, and c = 17. To solve it, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Substitute the values:

x = (20 ± √((-20)^2 - 4 * 2 * 17)) / 2 * 2

x = (20 ± √(400 - 136)) / 4

x = (20 ± √264) / 4

Now, we have two possible solutions:

1. x = (20 + √264) / 4
2. x = (20 - √264) / 4

Since the value inside the square root (264) is greater than zero, both solutions are real.

Next, let's simplify the expression: log7(28) - log7(4) + log7(49)

Using logarithm properties:

log_b(x) - log_b(y) = log_b(x/y)

log7(28) - log7(4) + log7(49) = log7(28/4) + log7(49)

Now, simplify further:

log7(7) + log7(49) = 1 + 2 = 3

So, the simplified expression is 3.

In summary, the solutions to the equation Log(1/2)(x^2 - 10x + 9) = 0 are:

1. x = (20 + √264) / 4
2. x = (20 - √264) / 4

And the simplified expression for log7(28) - log7(4) + log7(49) is 3.