Question

3) ㏒4(x+3)+㏒2(x+3)+㏒8(x+3)=1 4) ㏒1,7(x-1)-㏒1,7(x+1)≥0 5) ㏒1/4 2/4×√2.

Answer 1

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Answer 2

To solve the logarithmic equations, we'll use logarithmic properties and rules. Let's tackle each equation one by one:

1. ㏒4(x+3) + ㏒2(x+3) + ㏒8(x+3) = 1

Step 1: Combine the logarithms using the product rule: ㏒(a) + ㏒(b) = ㏒(a * b). ㏒4(x+3) + ㏒2(x+3) + ㏒8(x+3) = ㏒(4 * 2 * 8)(x+3) = ㏒64(x+3)

Step 2: Use the rule ㏒(a^b) = b * ㏒(a). ㏒64(x+3) = 1 can be rewritten as ㏒(x+3) = 1/㏒64.

Step 3: Evaluate the logarithm on the right side. ㏒(x+3) = 1/㏒64 ≈ 0.0931

Step 4: Use the definition of a logarithm to solve for x+3. x + 3 = 10^(1/㏒64) ≈ 1.197

Step 5: Solve for x. x ≈ 1.197 - 3 ≈ -1.803

2. ㏒1,7(x-1) - ㏒1,7(x+1) ≥ 0

Step 1: Use the quotient rule of logarithms: ㏒(a) - ㏒(b) = ㏒(a/b). ㏒1,7(x-1) - ㏒1,7(x+1) = ㏒1,7[(x-1)/(x+1)]

Step 2: Set the expression inside the logarithm greater than 1 since ㏒(a) is positive when a > 1. (x-1)/(x+1) > 1

Step 3: Solve the inequality for x. x - 1 > x + 1 -1 > 1 (This is not true for any value of x.)

Since the inequality is not true for any value of x, there are no solutions for this equation.

3. ㏒1/4(2/4) * √2

Step 1: Simplify the expression inside the logarithm and then take the logarithm. ㏒(1/4 * 1/2) * √2 = ㏒(1/8) * √2

Step 2: Use the property ㏒(a * b) = ㏒(a) + ㏒(b) to separate the expression. ㏒(1/8) * √2 = ㏒(1/8) + ㏒(√2)

Step 3: Evaluate the logarithms. ㏒(1/8) ≈ -3 ㏒(√2) ≈ 0.5

Step 4: Add the results together. ㏒(1/8) + ㏒(√2) ≈ -3 + 0.5 ≈ -2.5

So, the simplified value of ㏒1/4(2/4) * √2 is approximately -2.5.