2 ^ x = 3 ^ (x + 1) solve for x
1 answer:
Answer:
-2.71
Step-by-step explanation:
This exponential equation can be solved by rewriting it so there is only one base, then using logarithms.
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<h3>solve for x</h3>2^x = 3^(x+1) . . . . . given
2^x = 3·3^x . . . . . . eliminate the constant in the exponent
(2/3)^x = 3 . . . . . . . divide by 3^x
x·log(2/3) = log(3) . . . . take logarithms
x = log(3)/log(2/3) ≈ -2.7095113
The value of x is about -2.71.
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<em>Alternate solution</em>
Taking logarithms transforms this exponential equation to a linear equation.
x·log(2) = (x +1)·log(3) . . . . take logs
x(log(2) -log(3)) = log(3) . . . . . . subtract x·log(3)
x = log(3)/(log(2) -log(3)) = log(3)/log(2/3) . . . . . same result as above
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<em>Additional comment</em>
In the attached, we have used a graphing calculator to find the x-intercept of the function f(x) = 0, where f(x) = 2^x -3^(x+1). This is the solution to the given equation.
The applicable rules of logarithms are ...
log(a^b) = b·log(a)
log(a/b) = log(a) -log(b)
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Step-by-step explanation:
We have got the lines :
Both lines intercept the x-axis in the point :
In all point from x-axis the y-component is equal to 0.
We replace the I point in the lines equations:
From the first equation :
From the second equation :
Then
Finally :
y = ax + b and y = cx + d have the same x-intercept ⇔ad=bc
It is given in the question that B is in between C and D.
Therefore
Substituting the given values of CB,BD and CD, we will get
Combining like terms, we will get
Now we need to move 4 to the right side by adding 4 to both sides, that is