how do i solve these?
9^(-x)=1/3
2^(2x)+ 2^(x) -12 = 0
(3/5)^x = 7 ^ (1-x)
9^(-x)=1/3
(3^2)^(-x) = 3^-1
3^(-2x) = 3^-1
so -2x = -1
x = 1/2
2^(2x)+ 2^(x) -12 = 0
(2^x)^2 + 2^x - 12 = 0
let y = 2^x
y^2 + y - 12 = 0
(y+4)(y-3) = 0
y = -4 or y = 3
2^x = -4, no solution , or
2^x = 3 --- x = log3/log2
last one:
take logs of both sides
(3/5)^x = 7 ^ (1-x)
log(3/5)^x = log7 ^ (1-x)
xlog(3/5) = (1-x)Log 7
xlog .6 = log 7 - xlog 7
xlog .6 + xlog 7 = log 7
x(log .6 + log 7) = log 7
x = log 7/(log.6 + log7)
1/6
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To solve these equations, we'll need to apply some algebraic techniques. Let's go through each equation step-by-step:
1. 9^(-x) = 1/3:
To start, let's rewrite 1/3 as a power of 9. We know that 1/3 is the same as 3^(-1), so we can rewrite the equation as:
9^(-x) = 3^(-1)
Next, we can rewrite both sides using the same base:
(3^2)^(-x) = 3^(-1)
Now, apply the property of exponents that states (a^b)^c = a^(b * c):
3^(-2x) = 3^(-1)
Since the bases are the same, we can conclude that:
-2x = -1
Now, isolate x by dividing both sides of the equation by -2:
x = (-1) / (-2)
Simplifying the right side gives us:
x = 1/2
2. 2^(2x) + 2^x - 12 = 0:
Notice that we have a quadratic equation in terms of 2^x. We can make a substitution to simplify it:
Let y = 2^x.
Now our equation becomes:
y^2 + y - 12 = 0
Factor the quadratic equation:
(y + 4)(y - 3) = 0
Set each factor equal to zero:
y + 4 = 0 --> y = -4
y - 3 = 0 --> y = 3
Substitute back y = 2^x:
2^x = -4 and 2^x = 3
However, we cannot have negative values for 2^x, so we discard the solution 2^x = -4.
Now solve 2^x = 3:
Take the logarithm (base 2) of both sides of the equation:
x = log2(3)
Use the change of base formula for logarithms:
x = log(3) / log(2)
Approximately, x = 1.585
3. (3/5)^x = 7^(1-x):
To compare the bases of this equation, we can write 7 as (7/1):
(3/5)^x = (7/1)^(1-x)
Now, we can rewrite both sides using the same base:
(3/5)^x = (3/5)^(4-2x)
Since the bases are the same, we can equate the exponents:
x = 4 - 2x
Combine like terms:
3x = 4
Solve for x by dividing both sides by 3:
x = 4/3
Hence, the solution is x = 4/3.