Question

1). Log1/5(4-3x)>или=2 2). Log2x*log4x*log8x*log16x=1/3log0,5x

Answer 1

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Answer 2

Let's solve the inequalities step by step:

1. Solve for the inequality log1/5(4-3x) >= 2:

To begin, remember that log_a(b) = c is equivalent to a^c = b. In this case, we have log1/5(4-3x) >= 2, so it can be rewritten as:

1/5^(4-3x) >= 2

Next, eliminate the base by converting 2 to its corresponding logarithmic form with base 1/5:

1/5^(4-3x) >= 1/5^2

Now, since the bases are the same (both 1/5), we can compare the exponents:

4 - 3x >= 2

Now, solve for x:

-3x >= 2 - 4 -3x >= -2

Divide both sides by -3 (since we are dividing by a negative number, the inequality flips):

x <= 2/3

So the solution for the inequality is x <= 2/3.

2. Solve for the equation log2x * log4x * log8x * log16x = (1/3) * log0.5x:

First, notice that log0.5x is equal to log2(0.5x), as log with base 2 is the common logarithm used in most mathematical notations.

Now, rewrite the equation:

log2x * log4x * log8x * log16x = (1/3) * log2(0.5x)

Using the logarithmic property that log_a(b) + log_a(c) = log_a(b * c), we can simplify the left side:

log2x * log4x * log8x * log16x = log2(2x) * log2(2^2) * log2(2^3) * log2(2^4)

Using the fact that log_a(a^n) = n, the equation becomes:

log2x * log4x * log8x * log16x = log2(2x) * 2 * 3 * 4

Now, simplify the right side:

log2x * log4x * log8x * log16x = 24 * log2(2x)

Now, the equation becomes:

24 * log2(2x) = (1/3) * log2(0.5x)

Next, isolate the logarithm term on one side:

log2(2x) = (1/3) * log2(0.5x) / 24

Using the logarithmic property log_a(b^c) = c * log_a(b):

log2(2x) = log2((0.5x)^(1/3)) / 24

Now, since log_a(a) = 1 for any base a, we have:

2x = (0.5x)^(1/3) / 24

Raise both sides to the power of 3 to eliminate the cube root:

(2x)^3 = ((0.5x)^(1/3) / 24)^3

Simplify:

8x^3 = (0.5x) / (24^3)

Now, solve for x:

8x^3 = (0.5x) / 13824

Multiply both sides by 13824 to eliminate the denominator:

110592x^3 = 0.5x

Now, bring all terms to one side of the equation:

110592x^3 - 0.5x = 0

At this point, it's challenging to find the exact solutions for x. You may need to use numerical methods or calculators to approximate the values. However, it is evident that x = 0 is one of the solutions. To find the other solutions, you might use methods like the Newton-Raphson method or graphical methods.