Question

Log4(4x+2)=2 log1/3(x^2=8x)= -2 4^x<1/4 log2(8-4x) lgx=lg18 - lg3 +lg5 - lg15

Answer 1

Log4(4x+2)=2
4x+2=4²
4x=16-2
4x=14
x=14/4=7/2=3.5

log1/3(x^2-8x)= -2
x²-8x=(1/3)^-2
x²-8x=9
x²-8x-9=0
D=16+9=25
x(1)=4+5=9
x(2)=4-5=-1

4^x<1/4
4^x<4^ -1
x<-1

lgx=lg18 - lg3 +lg5 - lg15
lgx=lg(18+5)/3+15
x=23/18

Answer 2

Let's solve each equation step by step:

1. Log4(4x+2) = 2

To solve this equation, we can rewrite it in exponential form:

4^(log4(4x+2)) = 4^2

4x + 2 = 16

4x = 14

x = 14/4

x = 7/2 or 3.5

Therefore, the solution to the equation is x = 7/2 or x = 3.5.

2. log1/3(x^2) - 8x = -2

First, let's rewrite the equation using the properties of logarithms:

log1/3(x^2) - log1/3(8x) = -2

Now, we can combine the logarithms using the quotient rule:

log1/3(x^2 / (8x)) = -2

Simplify the expression inside the logarithm:

log1/3(x / 8) = -2

Next, we can rewrite the equation in exponential form:

1/3^(-2) = x / 8

3^2 = 8x

9 = 8x

x = 9/8 or 1.125

Therefore, the solution to the equation is x = 9/8 or x = 1.125.

3. 4^x < 1/4

To solve this inequality, we can rewrite 1/4 as a power of 4:

4^x < 4^(-1)

Since the bases are the same, we can compare the exponents:

x < -1

Therefore, the solution to the inequality is x < -1.

4. log2(8 - 4x) - lg(x) = lg(18) - lg(3) + lg(5) - lg(15)

Assuming "lg" represents the logarithm with base 10, we can simplify the equation:

log2(8 - 4x) - log10(x) = log10(18) - log10(3) + log10(5) - log10(15)

Apply the logarithmic properties:

log2(8 - 4x) - log10(x) = log10(18/3 * 5/15)

Simplify the fraction inside the logarithm:

log2(8 - 4x) - log10(x) = log10(3 * 1/3)

log2(8 - 4x) - log10(x) = log10(1)

Now, we can combine the logarithms:

log2((8 - 4x)/x) = 0

The logarithm with base 2 equals 0 if and only if the argument equals 1:

(8 - 4x)/x = 1

Solve for x:

8 - 4x = x

8 = 5x

x = 8/5 or 1.6

Therefore, the solution to the equation is x = 8/5 or x = 1.6.