Question

1) log6 (- 3x + 12) =2* 2)log5(-4x+5)= -1*

Answer 1

1) log₆ ( - 3x + 12 ) = 2

log₆ ( - 3x + 12 ) = 2, x < 4

- 3x + 12 = 6²

- 3x + 12 = 36

- 3x = 36 - 12

- 3x = 24

x = - 8

ответ: x = - 8

2) log₆ ( -4x + 5 )= - 1

log₆ ( -4x + 5 )= - 1, x < $\frac{5}{4}$

-4x + 5 = 6 ⁻¹

-4x + 5 = $\frac{1}{6}$

-4x = $\frac{1}{6}$ - 5

-4x = - $\frac{29}{6}$

x = $\frac{29}{24}$

x = 1 $\frac{5}{24}$

ответ: 1 $\frac{5}{24}$

Answer 2

To solve these logarithmic equations, you need to isolate the variable (x) by using the properties of logarithms. Let's solve them step by step:

1. log₆(-3x + 12) = 2

First, rewrite the equation in exponential form:

6^2 = -3x + 12

Now, solve for x:

36 = -3x + 12

Subtract 12 from both sides:

36 - 12 = -3x

24 = -3x

Now, divide both sides by -3 to isolate x:

x = -24 / -3 x = 8

So, the solution to the first equation is x = 8.

2. log₅(-4x + 5) = -1

Rewrite the equation in exponential form:

5^(-1) = -4x + 5

Since 5^(-1) is the same as 1/5, we have:

1/5 = -4x + 5

Subtract 5 from both sides:

1/5 - 5 = -4x

To subtract 5 from 1/5, you need to make them have a common denominator:

1/5 - 25/5 = -4x

Now, combine the fractions:

-24/5 = -4x

To isolate x, divide both sides by -4:

x = (-24/5) / (-4)

When you divide by a fraction, it's equivalent to multiplying by its reciprocal:

x = (-24/5) * (-1/4)

x = (24/5) * (1/4)

Now, multiply the numerators and denominators:

x = (24 * 1) / (5 * 4)

x = 24 / 20

Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 4:

x = 6/5

So, the solution to the second equation is x = 6/5.