So, the value of x is 1.19
The question is an exponential equation
<h3>What is an exponential equation?</h3>
An exponential equation is a mathematical expression between two quantities in which one variable is raised to a power of the other variable.
<h3>How to find x?</h3>
Since
,
Dividing through by 4ˣ, we have
![\frac{4^{x} }{4^{x} } + \frac{6^{x} }{4^{x} } = \frac{9^{x} }{4^{x} } \\1 + (\frac{6}{4})^{x} } = (\frac{9}{4})^{x} } \\1 + (\frac{3}{2})^{x} } = (\frac{3^{2} }{2^{2} })^{x} } \\1 + (\frac{3}{2})^{x} } = (\frac{3}{2})^{2x} }](https://tex.z-dn.net/?f=%5Cfrac%7B4%5E%7Bx%7D%20%7D%7B4%5E%7Bx%7D%20%7D%20%2B%20%5Cfrac%7B6%5E%7Bx%7D%20%7D%7B4%5E%7Bx%7D%20%7D%20%20%3D%20%5Cfrac%7B9%5E%7Bx%7D%20%7D%7B4%5E%7Bx%7D%20%7D%20%5C%5C1%20%2B%20%28%5Cfrac%7B6%7D%7B4%7D%29%5E%7Bx%7D%20%7D%20%20%3D%20%28%5Cfrac%7B9%7D%7B4%7D%29%5E%7Bx%7D%20%7D%20%5C%5C1%20%2B%20%28%5Cfrac%7B3%7D%7B2%7D%29%5E%7Bx%7D%20%7D%20%20%3D%20%28%5Cfrac%7B3%5E%7B2%7D%20%7D%7B2%5E%7B2%7D%20%7D%29%5E%7Bx%7D%20%7D%20%5C%5C1%20%2B%20%28%5Cfrac%7B3%7D%7B2%7D%29%5E%7Bx%7D%20%7D%20%20%3D%20%28%5Cfrac%7B3%7D%7B2%7D%29%5E%7B2x%7D%20%7D)
Let y = (3/2)ˣ
So,
1 + y = y²
y² - y - 1 = 0
Using the quadratic formula to find y,
![y = \frac{-b +/- \sqrt{b^{2} - 4ac} }{2a}](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-b%20%2B%2F-%20%5Csqrt%7Bb%5E%7B2%7D%20-%204ac%7D%20%7D%7B2a%7D)
where a = 1 b = -1 and c = -1
Substituting the values of the variables into the equation, we have
![y = \frac{-(-1) +/- \sqrt{(-1)^{2} - 4\times 1 \times (-1)} }{2\times 1}\\= \frac{1 +/- \sqrt{1 + 4} }{2}\\= \frac{1 +/- \sqrt{5} }{2}\\= \frac{1 - \sqrt{5} }{2} or \frac{1 + \sqrt{5} }{2}\\= \frac{1 - 2.236}{2} or \frac{1 + 2.236}{2}\\= \frac{- 1.236}{2} or \frac{3.236}{2}\\= -0.618 or 1.618](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7B-%28-1%29%20%2B%2F-%20%5Csqrt%7B%28-1%29%5E%7B2%7D%20-%204%5Ctimes%201%20%5Ctimes%20%28-1%29%7D%20%7D%7B2%5Ctimes%201%7D%5C%5C%3D%20%5Cfrac%7B1%20%2B%2F-%20%5Csqrt%7B1%20%2B%204%7D%20%7D%7B2%7D%5C%5C%3D%20%5Cfrac%7B1%20%2B%2F-%20%5Csqrt%7B5%7D%20%7D%7B2%7D%5C%5C%3D%20%5Cfrac%7B1%20-%20%5Csqrt%7B5%7D%20%7D%7B2%7D%20or%20%20%5Cfrac%7B1%20%2B%20%5Csqrt%7B5%7D%20%7D%7B2%7D%5C%5C%3D%20%5Cfrac%7B1%20-%202.236%7D%7B2%7D%20or%20%20%5Cfrac%7B1%20%2B%202.236%7D%7B2%7D%5C%5C%3D%20%5Cfrac%7B-%201.236%7D%7B2%7D%20or%20%20%5Cfrac%7B3.236%7D%7B2%7D%5C%5C%3D%20-0.618%20or%201.618)
Since y = (3/2)ˣ
Takung logarithm of both sides, we have
㏒y = ㏒(3/2)ˣ
㏒y = x㏒(3/2)
x = ㏒y/㏒(3/2)
x = ㏒y/㏒1.5
Since we do not have logarithm of a negative number, we use y = 1.618.
So, x = ㏒y/㏒1.5
x = ㏒1.618/㏒1.5
x = 0.2090/0.1761
x = 1.19
So, the value of x is 1.19
Learn more about exponential equation here:
brainly.com/question/11832081
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