Question

1) ㏒₀,₂16-5㏒₀,₂₅8 2) ㏒₂(7-8x)=2 3) ㏒(5-x)-㏒(1-x)=㏒2 4) ㏒²₄(x-3)-㏒₄(x-3)-2=0

Answer 1

Смотри приложенное решение

Answer 2

To solve these logarithmic expressions and equations, I'll provide step-by-step solutions for each one:

1. ㏒₀,₂16 - 5㏒₀,₂₅8:

Here, we are dealing with logarithms of base 2.

First, let's simplify the expression: ㏒₀,₂16 = log base 2 of 16 ㏒₀,₂258 = log base 2 of 258

Now, you can use the properties of logarithms to simplify the expression:

㏒₀,₂16 - 5㏒₀,₂258 = log₂16 - 5log₂258

Use the property: log₂(ab) = log₂a + log₂b

㏒₀,₂16 - 5㏒₀,₂258 = log₂(16 * 258) - log₂(258^5)

Now, calculate the values inside the logarithms:

㏒₀,₂16 - 5㏒₀,₂258 = log₂(4128) - log₂(258^5)

Next, use the property: log₂(a^n) = n * log₂(a)

㏒₀,₂16 - 5㏒₀,₂258 = log₂(4128) - 5 * log₂(258)

You can approximate the values using a calculator:

㏒₀,₂16 - 5㏒₀,₂258 ≈ 12 - 5 * 8.003 = 12 - 40.015 ≈ -28.015

So, ㏒₀,₂16 - 5㏒₀,₂258 is approximately -28.015.

2. ㏒₂(7 - 8x) = 2:

In this equation, the logarithm is of base 2.

To solve for x, we need to isolate the logarithm expression:

Step 1: Rewrite the equation in exponential form: 2^2 = 7 - 8x

Step 2: Simplify the exponential expression: 4 = 7 - 8x

Step 3: Isolate x by moving the constant term to the other side: 8x = 7 - 4

Step 4: Divide by 8 to solve for x: x = (7 - 4) / 8 x = 3 / 8

So, the solution for the equation is x = 3/8.

3. ㏒(5 - x) - ㏒(1 - x) = ㏒2:

In this equation, the base of the logarithm is not specified, so we assume it's a common logarithm (base 10).

Step 1: Combine the logarithms on the left side using the quotient rule of logarithms: ㏒[(5 - x) / (1 - x)] = ㏒2

Step 2: Convert the equation to exponential form: 10^(㏒[(5 - x) / (1 - x)]) = 10^(㏒2)

Step 3: Simplify the equation: (5 - x) / (1 - x) = 2

Step 4: Solve for x: Cross-multiply to eliminate the fraction: 5 - x = 2(1 - x)

Distribute 2 on the right side: 5 - x = 2 - 2x

Step 5: Isolate x on one side: -x + 2x = 5 - 2

Simplify: x = 3

So, the solution for the equation is x = 3.

4. ㏒²₄(x - 3) - ㏒₄(x - 3) - 2 = 0:

In this equation, we are dealing with logarithms of base 24 and base 4.

Step 1: Combine the logarithms on the left side using the quotient rule of logarithms: ㏒(x - 3) / ㏒₂₄ - ㏒(x - 3) / ㏒₄ - 2 = 0

Step 2: Find common logarithms (base 10) for both ㏒₂₄ and ㏒₄: ㏒(x - 3) / ㏒10 - ㏒(x - 3) / ㏒2 - 2 = 0

Step 3: Simplify the equation using properties of logarithms: (㏒(x - 3) - ㏒(x - 3) * ㏒10 / ㏒2) - 2 = 0

Step 4: Cancel out the common logarithm: ㏒(x - 3) * (1 - ㏒10 / ㏒2) - 2 = 0

Step 5: Calculate the value inside the parenthesis: 1 - ㏒10 / ㏒2 ≈ 0.68

Step 6: Isolate ㏒(x - 3) by moving the constant term to the other side: ㏒(x - 3) ≈ 2 / 0.68

Step 7: Solve for x by converting the equation to exponential form: x - 3 ≈ 10^(2 / 0.68)

Step 8: Simplify: x - 3 ≈ 10^(2.9412)

Step 9: Solve for x: x ≈ 10^(2.9412) + 3

Now, use a calculator to evaluate the expression:

x ≈ 880.83 + 3 x ≈ 883.83

So, the approximate solution for the equation is x ≈ 883.83.