1.Log10(x²-12x+36)=2
2.log4 log3 log2 x=0
3.log3 [log9x+1/2+9x]=2x
4.2log4(4-4)=4-log2(-2-x)
1. by definition of logs
x^2 - 12x + 36 = 10^2
(x-6)^2 = 100
x-6 = ±√100 = ±10
x = 6 ± 10 = 16 or -4
2. you have the product of 3 numbers = 0
so one of them is zero
but log4 and log2 are non-zero real numbers, so
log2x must be 0
that is, x = 1
3.not clear about your typing here.
my first reaction is to see:
3^(2x) = log9x + 1/2 + 9x
-- that would be a horrible equation to solve
4. same thing, I have doubts about your typing.
Why have (4-4) , that would be 0 and log 0 is undefined.
To find the solutions to these logarithmic equations, we can use properties of logarithms and algebraic manipulations. Let's solve each of these equations step by step:
1. Log10(x^2 - 12x + 36) = 2
First, we can rewrite the equation using an exponential form of logarithm:
10^2 = x^2 - 12x + 36
Simplifying it further:
100 = x^2 - 12x + 36
Rearrange:
x^2 - 12x - 64 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Factoring is the simplest method in this case:
(x - 8)(x - 8) = 0
This gives us x = 8 as the only solution for this equation.
2. log4(log3(log2(x))) = 0
To start, we need to apply exponentiation to both sides of the equation to eliminate the logarithms.
4^0 = log3(log2(x))
Simplifying,
1 = log3(log2(x))
Next, we want to eliminate the logarithm on the left side.
3^1 = log2(x)
This simplifies to:
3 = log2(x)
Applying exponentiation again:
2^3 = x
Hence, x = 8 is the solution to this equation.
3. log3(log9x + (1/2) + 9x) = 2x
First, let's simplify the inner part of the logarithm by combining terms:
log3(log9x + (1/2) + 9x) = 2x
log3((18x + (1/2)) + 9x) = 2x
log3((27x + 1/2) = 2x
To eliminate the logarithm, we apply exponentiation:
3^(2x) = 27x + 1/2
Now, we can manipulate the equation further:
9^x = 27x + 1/2
(3^2)^x = 27x + 0.5
3^(2x) = 27x + 0.5
At this point, we need to solve this equation either by numerical approximation or graphical methods since it cannot be easily simplified.
4. 2log4(4-4) = 4 - log2(-2-x)
Starting with the left side of the equation, we can simplify it:
2log4(0) = 4 - log2(-2-x)
Since log4(0) is undefined, the left side of the equation is not defined.
This means the equation does not have a solution.