Question

Log5 (x-3)=5 Розв'язати нерівність 1. (0,4)^х>0,16 2. log3 (x-2)-2 log3•2<0

Answer 1

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Answer 2

To solve the inequalities, we'll work through them step by step:

1. (0,4)^x > 0.16

First, let's find the base of the logarithm that was used to represent the left side of the inequality. Since the expression (0,4)^x does not explicitly mention the base, we can assume it's 10 (common logarithm) or e (natural logarithm). In this case, let's assume it's 10.

(0.4)^x > 0.16

Now, we can use logarithms to solve for x:

Take the logarithm of both sides using base 10 (logarithms of different bases are proportional to each other):

log(0.4)^x > log(0.16)

Apply the power rule of logarithms (log(a^b) = b * log(a)):

x * log(0.4) > log(0.16)

Now, divide both sides by log(0.4). Note that log(0.4) is negative since 0 < 0.4 < 1:

x < log(0.16) / log(0.4)

Using a calculator, we find:

x < -0.79248

So, the solution for the inequality is x < -0.79248.

2. log3 (x-2) - 2 log3•2 < 0

First, let's simplify the expression on the left side:

log3(x - 2) - 2log3(2) < 0

Now, we can combine the logarithms:

log3[(x - 2) / 2^2] < 0

Simplify further:

log3[(x - 2) / 4] < 0

Next, let's rewrite the inequality in exponential form:

3^0 > (x - 2) / 4

Since 3^0 = 1, we have:

1 > (x - 2) / 4

Now, multiply both sides by 4:

4 > x - 2

Add 2 to both sides:

6 > x

So, the solution for the inequality is x < 6.

In summary:

1. The solution for the inequality (0.4)^x > 0.16 is x < -0.79248.
2. The solution for the inequality log3 (x-2) - 2 log3•2 < 0 is x < 6.