Solve Exponent Problem: 3^x=4^y=12^z

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In summary, the equation 3^x=4^y=12^z is a logarithmic equation where the values of x, y, and z can vary based on given parameters. To solve for these variables, one must use logarithms. The variables are related by a common exponent, meaning that if one value is known, the others can be solved for using logarithms. This method is necessary, as it is not possible to solve for x, y, and z without using logarithms. Some real-life applications of solving exponent problems include various scientific fields, financial planning, and population growth predictions.
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Given \(\displaystyle 3^x=4^y =12^z\) show that \(\displaystyle z=\frac{xy}{x+y}\)

I've take logs on both sides and find myself going in circles, any hints?
 
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\(\displaystyle 3^{xy/(x+y)}=12^{zy/(x+y)}\)

Now try using the fact that $3$ and $4$ are factors of $12$ and that $4^y=3^x$.
 

FAQ: Solve Exponent Problem: 3^x=4^y=12^z

What is the value of x, y, and z in the equation 3^x=4^y=12^z?

The equation 3^x=4^y=12^z is known as a logarithmic equation. In order to solve for x, y, and z, you will need to use logarithms. The value of x, y, and z can vary based on the given parameters.

How do you solve an exponent problem like 3^x=4^y=12^z?

To solve an exponent problem like 3^x=4^y=12^z, you will need to use logarithms. Specifically, you will need to take the logarithm of both sides of the equation. This will allow you to solve for the unknown variables x, y, and z.

What is the relationship between x, y, and z in the equation 3^x=4^y=12^z?

In this equation, x, y, and z are related by a common exponent. This means that if you know the value of any one of the variables, you can use logarithms to solve for the other two variables.

Can you solve for x, y, and z without using logarithms?

No, it is not possible to solve for x, y, and z in the equation 3^x=4^y=12^z without using logarithms. This is because the variables are related by a common exponent, and in order to solve for them, you will need to use logarithms to isolate each variable.

What are some real-life applications of solving exponent problems?

Solving exponent problems can be useful in a variety of scientific fields, such as physics, chemistry, and engineering. It can also be used in financial planning, population growth predictions, and computer programming. Essentially, any situation where variables are related by a common exponent can be solved using logarithms.

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