Question

Can someone solve this quick

Answer 1

x = 1/2

Explanation:

4^x * 3^2x = 6

Now 3^2 = 9, so:

4^x * 9^x = 6

36^x = 6

6^2x = 6^1

2x = 1

x = 1/2

Answer 2

x = 1/2

Explanation:

Given:

`4^x * 3^(2x)=6`

To solve the given equation for 'x,' we'll need to use logarithms and some basic algebraic manipulation. Let's break down the process step-by-step:

(1) Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

`\Longrightarrow \ln(4^x * 3^(2x))=\ln(6)`

(2) Use the properties of natural logarithms to expand the L.H.S.

`\Longrightarrow \ln(4^x) + \ln (3^(2x))=\ln(6)\\\\\\\\\Longrightarrow x\ln(4) + 2x\ln (3)=\ln(6)`

(3) Factor out an 'x' on the L.H.S.

`\Longrightarrow x(\ln(4) + 2\ln (3))=\ln(6)`

(4) Divide each side of the equation by 'ln(4) + 2ln(3)'.

`\Longrightarrow x=(\ln(6))/(\ln(4) + 2\ln (3))`

(5) We can use the properties of natural logarithms to simplify the answer.

`\Longrightarrow x=(\ln(6))/(2\ln (6))\\\\\\\\\therefore \boxed{\boxed{x=(1)/(2) }}`

Thus, the solution is found.

`\hrulefill`

Additional Information:

Properties of Natural Logarithms: Natural logarithms have several properties that are important in various mathematical and scientific applications. Here's a list of some properties of natural logarithms:

`\boxed{\left\begin{array}{ccc}\text{\underline{Properties of Natural Logs:}}\\\\\ln(ab)=\ln(a)+\ln(b)\\\\\ln\Big((a)/(b)\Big)=\ln(a)-\ln(b)\\\\\ln(m^p)=p\ln(m)\\\\\ln(e^x)=x; \ \text{for} \ x > 0\\\\\text{if} \ e^x=e^y, \ \text{then} \ x=y\\\\\text{if} \ \ln(x)=\ln(y), \ \text{then} \ x=y\end{array}\right }`